On depth zero L-packets for classical groups

By computing reducibility points of parabolically induced representations, we construct, to within at most two unramified quadratic characters, the Langlands parameter of an arbitrary depth zero irreducible cuspidal representation $\pi$ of a classical group (which may be not-quasi-split) over a nonarchimedean local field of odd residual characteristic. From this, we can explicitly describe all the irreducible cuspidal representations in the union of one, two, or four L-packets, containing $\pi$. These results generalize the work of DeBacker-Reeder (in the case of classical groups) from regular to arbitrary tame Langlands parameters.


Introduction
The representation theory of p-adic groups has largely been motivated, over the last half-century, by the Langlands conjectures, seeking an understanding of the absolute Galois (or Weil) groups of local and global fields. Many parts of the local conjectures are now theorems, notably for representations of GL n [20,21], of SL n [17,22] and, more recently, of classical groups [2,43,29].
At the same time as having local Langlands correspondences, one would like to be able to use them to translate fine arithmetical data between representations of p-adic groups and representations of the local Weil group. To this end, one seeks to make the correspondence explicit/effective. For GL n , this has been the subject of a series of papers by Bushnell-Henniart [7,8,9,11]; for other groups, work has concentrated on regular depth zero irreducible cuspidal representations [14,24,27] and epipelagic irreducible cuspidal representations [46,19,26,47,25,10], with the most general work by Kaletha [28] on regular cuspidal representations.
In this spirit, we look here at depth zero irreducible cuspidal representations of a classical group G -by which we mean a symplectic, (special) orthogonal, or unitary group, which may be non-quasi-split -over a nonarchimedean locally compact local field of odd residual characteristic (this is the only restriction on the field). When these representations are also regular (more precisely, the corresponding Langlands parameter is tame regular semisimple in general position), these have already been considered, for more general groups but with some conditions on the field, by DeBacker-Reeder [14] and Kaletha [24,27]; however, our approach here is different, and allows us to treat all depth zero irreducible cuspidal representations. Thus our work, and methods, are complementary to those of [28].
Given a Langlands parameter for G, the Langlands correspondence should determine an L-packet of irreducible smooth complex representations of G. These representations should share many properties; for example, they should have all the same L-functions, at least where these have been defined. Since, by the results of Shahidi [48], poles of L-functions correspond to reducibility points of parabolic induction, we detect representations in the same L-packet by computing these reducibility points, and this does not require, for example, genericity of the representation.
For now, we are not able completely to compute reducibility points, but only up to twist by a certain unramified character (see below for more details). However, an evendimensional irreducible tame representation of the Weil group is symplectic if and only if this unramified twist is orthogonal; thus, using the Langlands correspondence, for example for symplectic groups, we can see which of the twists must occur, and the only ambiguity is in the reducibility points for quadratic characters of GL 1 . In any case, we are able to recover the irreducible cuspidal representations in the union of either one, two or four L-packets.
This paper can be regarded as a first step in a programme to treat all discrete series representations of classical groups -see [6] for the case of arbitrary irreducible cuspidal representations of symplectic groups. Depth zero is the base case, since general irreducible cuspidal representations are built from a "wild part" and a depth zero part (see [51]). In the depth zero case, we avoid the complication of wild ramification; on the other hand, the geometric complications arise essentially from the depth zero part so that the results and techniques here already resolve many difficulties for the general case. Now let us state our results more carefully; although we have interpreted them above via the Langlands correspondence, they are in fact results on the automorphic side. Let F{F o be an extension of degree at most two of nonarchimedean local fields of odd residual characteristic, and let G be (the group of rational points of) a symplectic, special orthogonal or unitary group over F o , the connected component of the group of isometries of an F{F o -hermitian space; this group may be non-quasi-split. We also write W F for the Weil group of F and p G for the complex dual group of G, acting naturally on a vector space of dimension N p G . In their classification of discrete series representations of (quasi-split) p-adic classical groups [42,37,39], Moeglin-Tadić use the notion of a Jordan set attached to an irreducible discrete series representation of G. For an irreducible cuspidal representation π of G, this can be described via the reducibility set Redpπq as follows.
We denote by A σ pFq the set of (equivalence classes of) self-dual irreducible cuspidal representations of some GL n pFq (see Section 3). For ρ P A σ pFq there is at most one real number s " s π pρq ě 0 such that the normalized parabolically induced representation Ind ρ| detp¨q| s F b π is reducible, where |¨| F is the normalized absolute value on F; when there is no such real number (which can happen only for even-dimensional special orthogonal groups and ρ a quadratic character of GL 1 pFq), we set s π pρq " 0. Then Redpπq " tpρ, mq : ρ P A σ pFq, m P N with 2s π pρq " m`1u .
It is expected (and in at least when G is quasi-split, known -see, for example, [41]) that the Jordan set should precisely predict the Langlands parameter ϕ : W FˆS L 2 pCq Ñ p G¸W F whose L-packet Π ϕ contains π, by ϕ " à pρ,mqPJordpπq where ϕ ρ is the (irreducible) representation of the Weil group W F corresponding to ρ via the Langlands correspondence for general linear groups, and st m is the m-dimensional irreducible representation of SL 2 pCq. In particular, writing n ρ for the unique natural number such that ρ is a representation of GL nρ pFq, we should have By (1.1), this equality is equivalent to where txu denotes the greatest integer not exceeding x. Note that almost all terms in this sum are zero since s ρ pπq ă 1 for all but finitely many ρ P A σ pFq.
Suppose now that the representation π is of depth zero; equivalently, the Langlands parameter is tame (i.e. trivial on restriction to the wild inertia subgroup of W F ). For clarity of exposition, we specialize temporarily to the case of a symplectic group G, in which case p G is a special orthogonal group with N p G odd. On the other hand, by a result of Blondel [3], there are self-dual irreducible cuspidal representations of GL n pFq only for n even or n " 1; in the latter case, we get the pair of unramified characters of order dividing two, and the pair of (tamely) ramified quadratic characters. Since N p G is odd, equation (1.2) implies that there is exactly one pair for which the multiplicities of the two characters in ϕ| WF have the same parity; we denote by ϕ 1 the Langlands parameter obtained from ϕ by exchanging the multiplicities of the two characters in this pair. (In particular, we have ϕ 1 " ϕ when the multiplicities are equal.) Theorem .
(i) Given a tame Langlands parameter ϕ for a symplectic group as above, there is an explicit description of the cuspidal representations in the union of L-packets Π ϕ Y Π ϕ 1 .
(ii) Conversely, given a depth-zero cuspidal irreducible representation π of a symplectic group as above, there is an explicit description of the pair tϕ, ϕ 1 u of tame Langlands parameters, such that π P Π ϕ Y Π ϕ 1 .
We remark that, in the situation of regular depth zero irreducible cuspidal representations, the multiplicities of the characters are all at most one, so that ϕ 1 " ϕ; thus we recover the description of the representations in an L-packet consisting solely of cuspidal representations from [14] in this case.
We return to the case of depth zero representations of a general classical group G and describe the result here, which is a reinterpretation of the theorem above in terms of the set Redpπq. More precisely, denote by rρs the inertial equivalence class of ρ P A σ pFq, that is, the set of unramified twists of ρ; note that rρs X A σ pFq " tρ, ρ 1 u consists of exactly two (inequivalent) representations. We have ρ 1 " ρχ, for χ an unramified character with ρχ 2 " ρ; since ρ has depth zero, the character χ has order 2n ρ .
Here, we compute the inertial reducibility multiset IRedpπq " ttprρs, mq : pρ, mq P Redpπquu. This is often in fact a set: since π has depth zero, the only inertial classes rρs which can occur with multiplicity are the quadratic characters of GL 1 pFq. Indeed, for ρ P A σ pFq of depth zero with n ρ ą 1 and ρ 1 its self-dual unramified twist, the exterior square Lfunction of exactly one of ρ, ρ 1 has a pole at s " 0, while for the other representation it is the symmetric square L-function which has a pole (the same comments apply to the Asai and twisted Asai L-functions when F{F o is quadratic); thus the parity of m, such that pρ, mq P Redpπq should be independent of π (i.e. depend only on ρ), and the parity will be the opposite of that for ρ 1 . Moreover, this means that by computing IRedpπq, we in fact know all elements of Redpπq apart from those associated to characters of GL 1 pFq of order at most two, where an ambiguity may remain. The results we prove here can be described by the following: Theorem . Let π be a depth zero irreducible cuspidal representation of G.
The multiset IRedpπq can be computed explicitly from the local data defining π as a compactly induced representation. (iii) The set of irreducible cuspidal representations π 1 of G with IRedpπq " IRedpπ 1 q can be described explicitly in terms of the local data defining π. Moreover, the number of such representations is the expected number in one, two or four Lpackets, this number depending again on the local data.
For a discussion of the expected number of irreducible cuspidal representations in an L-packet, see the beginning of Section 9. One also needs to take care with this in the case of even orthogonal groups (see Example 9.6).
The computation of reducibility points required for this theorem is achieved using Bushnell-Kutzko's theory of covers [12], together with results of Blondel [4], which translate the problem to the computation of parameters in the Hecke algebra of a cover (see . These covers were constructed in [36], and the parameters can be computed using Morris's explicit description of depth zero irreducible cuspidal representations (see Section 3), together with results of Lusztig on representations of finite reductive groups (see Section 7). However, care must be taken since the finite reductive groups which occur do not, in general, have connected centre. There is particular difficulty for (even-dimensional) special orthogonal groups and the results we obtain here may be of independent interest; in particular, we compute when an irreducible cuspidal representation of an even-dimensional special orthogonal group over a finite field extends to the full orthogonal group (see Proposition 7.9), generalizing results of Lusztig and Waldspurger. All these ingredients are then put together to prove the theorem in Sections 8 and 9; the latter includes some illustrative examples.
Acknowledgements. The research of SS was supported by EPSRC grants EP/G001480/1 and EP/H00534X/1. He would like to thank Meinolf Geck, and particularly Marc Cabanes for his patience in explaining Deligne-Lusztig theory -any remaining mistakes are entirely the authors'. He would also like to thank Corinne Blondel and Guy Henniart for their patience in waiting for this paper to get written up.

Notation and background
We fix some notation for the rest of the paper (with the exception of Section 7, whose notation is independent). Let F o be a locally compact nonarchimedean local field of odd residual characteristic p, and let F{F o be an extension of degree at most 2. We write σ : λ Þ Ñ λ for the generator of the Galois group of F{F o . For E any field containing F o , we write o E for its ring of integers, p E for its maximal ideal, and k E " o E {p E for its residue field, of cardinality q E ; in particular we abbreviate q " q F . We also abbreviate o o " o Fo etc. We also fix a uniformizer ̟ F of F such that ̟ F "´̟ F if F {F o is quadratic ramified, and ̟ F " ̟ F otherwise, and write N F{Fo : FˆÑ Fô for the norm map, which is given by λ Þ Ñ λλ if rF : F o s " 2.
We fix a sign ε "˘1, and let pV, hq be a nondegenerate F{F o -ε-hermitian space of Witt index N and dimension 2N`N an ; thus V is an F-vector space, the form h satisfies hpλv, µwq " λµhpv, wq " ελµhpw, vq, for v, w P V, λ, µ P F, and we have a Witt decomposition with dim F V˘" N and dim F V an " N an , such that the restriction of h to V˘is totally isotropic, while its restriction h an to V an is anisotropic. We denote by H " H´' H`the hyperbolic plane; that is, H˘is a 1-dimensional F-vector space with basis e˘and H is equipped with the form h H given by h H pλ´e´`λ`e`, µ´e´`µ`e`q " λ´µ``ελ`µ´, for λ˘, µ˘P F.
Thus the restriction of h to V´' V`is (isometric to) an orthogonal direct sum of N copies of H. We choose a Witt basis for V, that is: e1 , . . . , eǸ a basis for V`, with dual basis e1 , . . . , eŃ for V´, and e an 1 , . . . , e an N an a basis V an with respect to which h an has diagonal Gram matrix. We order this basis eŃ , . . . , e1 , e an 1 , . . . , e an N an , e1 , . . . , eǸ . For n ě 0, we denote by nH the orthogonal direct sum of n copies of H, and put V n " V ' nH with the form h n " h ' h H '¨¨¨' h H , so that the decomposition above is orthogonal and we have a Witt decomposition with Vn " V˘' nH˘.
Thus pV n : n ě 0q is a Witt tower over V 0 " V. Writing eN`i for the image in V n of eȋ n the i th copy of H, the space V n has the ordered Witt basis eŃ`n, . . . , e1 , e an 1 , . . . , e an N an , e1 , . . . , eǸ`n. For n ě 0, we put Gǹ " UpV n q, the group of F o -rational points of the reductive algebraic group over F o determined by pV n , h n q, so that Gǹ " tg P Aut F pV n q : h n pgv, gwq " h n pv, wq for all v, w, P Vu; thus Gǹ is (the group of points of) a unitary, symplectic or (full) orthogonal group. We also put G n " UpV n q o , the group of F o -rational points of the connected component, so that G n " tg P Gǹ : N F{Fo det F pgq " 1u; thus G n " Gǹ unless Gǹ is an orthogonal group (so F " F o and ε " 1), in which case G n is the special orthogonal group, of index 2 in Gǹ . We will abbreviate G " G 0 and G`" G0 . The stabilizer in G n of the decomposition is a Levi subgroup M n of G n , which is standard with respect to the chosen Witt basis, and we have an isomorphism M n » GL n pFqˆG given by g Þ Ñ pg| nH´, g| V q; moreover, the stabilizer of the subspace nH´is a standard parabolic subgroup P n of G n , with Levi component M n . Thus, writing elements of G n as matrices with respect to the Witt basis, the group P n is block upper triangular and M n is block diagonal.
We end this section with a description of the maximal parahoric subgroups of G and of their reductive quotients (see also [44]). For L an o F -lattice in V, we denote by L # the dual lattice L # " tv P L : hpv, Lq Ď p F u. We say that L is almost self-dual if in that case, the stabilizer J " J L in G of L is a maximal compact subgroup of G, and every maximal compact subgroup arises in this way for a unique self-dual lattice L. We write J 1 for the pro-unipotent radical of J, that is the subgroup consisting of those elements g which induce the identity map on the k F -vector spaces V p1q :" L{L # and V p2q : is quadratic ramified, and ε-hermitian otherwise, by our choice of uniformizer. Thus we get an induced map J Ñ UpV p1q qˆUpV p2q q, with kernel J 1 , and hence the quotient G " G L " J{J 1 is naturally a subgroup of the finite reductive group UpV p1q qˆUpV p2q q. In fact, G identifies with the subgroup pg 1 , g 2 q P UpV p1q qˆUpV p2q q : N kF{ko pdet kF pg 1 q det kF pg 2 qq " 1 ( , which has connected component G o " UpV p1q q oˆU pV p2q q o . We denote by J o " J o L the inverse image in J of G o ; this is a parahoric subgroup of G and J is its normalizer in G. It is not always a maximal parahoric subgroup of G (it is so if and only if neither factor UpV piq q o is a two-dimensional special orthogonal group) but every maximal parahoric subgroup does arise in this way. If either F{F o is quadratic ramified and the orthogonal space among V p1q , V p2q is non-zero, or F " F o , ε " 1 and both V p1q , V p2q are non-zero, then J o has index 2 in J; otherwise we have J " J o .
Restricting first to the case of the hermitian space pV an , h an q, there is a unique almost self-dual lattice L an in V an , and the corresponding group G an " UpV an q o is compact and normalizes the unique (maximal) parahoric subgroup J o an " J o Lan , with connected component G o an . We set V ‚ If F " F o and ε "´1 then N an " 0 so G o an is trivial.
with N an i ď 2 and only one N an i non-zero.
Returning to the general case of the space pV, hq, the standard almost self-dual lattices are those of the following form: for 0 ď N 1 , N 2 with N 1`N2 " N , set where L an is the unique almost self-dual lattice in V an . We write J N1,N2 for the stabilizer of L N1,N2 and J o N1,N2 for the corresponding parahoric subgroup. Every almost self-dual lattice has the form gL N1,N2 , for some g P G and a unique standard lattice L N1,N2 ; thus every maximal compact (respectively, maximal parahoric) subgroup is conjugate to a unique standard one J N1,N2 (respectively, J o N1,N2 ). The choice of Witt basis and the forms on V p1q , V p2q then give us the following identifications for the connected reductive Writing H for hyperbolic space over k F , we can unify these by writing an piq q o , for i " 1, 2. We note that J o N1,N2 is a maximal parahoric subgroup except where one of the factors here is SOp1, 1, k F q but G is not itself a 2-dimensional special orthogonal group; that is, in the following cases: ‚ F " F o , ε " 1, pN, N an q ‰ p1, 0q, with pN i , N an i q " p1, 0q, for i " 1 or 2; ‚ F{F o is quadratic ramified, N an " 0 and pε, N 1 q " p1, 1q or pε, N 2 q " p´1, 1q.

Depth zero cuspidal representations
In this section, we recall the classification of the depth zero irreducible cuspidal representations of GL n pFq and of the classical group G, beginning with the former.
We write A n pFq for the set of equivalence classes of irreducible cuspidal representations of GL n pFq and put ApFq " Ť ně1 A n pFq. We will abuse notation by writing ρ P ApFq to mean ρ is an irreducible cuspidal representation of some GL n pFq, where n " n ρ is of course uniquely determined by ρ. For ρ P A n pFq, we denote by ρ σ the representation where σpgq denotes the matrix obtained by applying the generator σ of GalpF{F o q to each entry, and g T denotes the transpose matrix. We say that ρ is self-dual if ρ σ » ρ, and write A σ n pFq for the set of equivalence classes of self-dual irreducible cuspidal representations of GL n pFq, and A σ pFq " Ť ně1 A σ n pFq for the set of equivalence classes of self-dual representations in ApFq.
We do not recall here the general notion of depth, only that a representation ρ P ApFq is said to be of depth zero if it has fixed vectors under the pro-unipotent radical of the maximal parahoric subgroup GL nρ po F q of GL nρ pFq. We denote by A r0s pFq the set of equivalence classes of depth zero representations in ApFq, and by A σ r0s pFq the set of equivalence classes of self-dual depth zero representations in ApFq. Any depth zero representation ρ P A n pFq can be written where J ρ " FˆGL n po F q is the normalizer of the maximal parahoric subgroup J ρ " GL n po F q of GL n pFq, and Λ ρ is an irreducible representation of J ρ whose restriction λ ρ " Λ ρ | Jρ is the inflation of an irreducible cuspidal representation τ ρ of the reductive quotient J ρ {J 1 ρ » GL n pk F q. Moreover, the (equivalence class of the) representation τ " τ ρ is uniquely determined by ρ. Further, ρ is self-dual if and only if τ is self-dual; that is, denoting again by σ the generator of Galpk F {k o q and by τ σ the representation τ σ pgq " τ ρ pσpg´1q T q, for g P GL n pk F q, we have τ σ » τ .
The (equivalence classes of) irreducible cuspidal representations τ of GL n pk F q were first classified by Green [18], and are parametrized by regular characters of the multiplicative group of the degree n extension of k F , or, equivalently (after making choices), by monic irreducible degree n polynomials P " P τ P k F rXs with P p0q ‰ 0. Writing σpP q for the polynomial obtained by applying σ to the coefficients of P , the representation τ σ then corresponds to the polynomial P σ pXq :" σpP p0qq´1X degpP q σpP qp1{Xq. Thus τ is self-dual if and only if P τ " P σ τ . If k F " k o , such polynomials exist if and only if n " 1 or n is even (see [1]); if k F {k o is quadratic, then such polynomials exist if and only if n is odd (see [30, §5.4]).
Similarly, we write ApGq for the set of equivalence classes of irreducible cuspidal representations of G, and A r0s pGq for the subset of equivalence classes of depth zero representations. Then, for π P A r0s pGq, we can write π " c-Ind G Jπ λ π , where J π " J N1,N2 is the (compact open) normalizer of a standard maximal parahoric subgroup J o π and λ π is an irreducible representation of J π whose restriction λ o π " λ π | J o π is a sum of conjugates (under J π ) of the inflation of an irreducible cuspidal representation τ π of the reductive quotient G o N1,N2 . By [45,52], the standard maximal parahoric subgroup J o π is uniquely determined by π, so that N 1 , N 2 here are determined by π, and the representation τ π is determined up to conjugacy by an element of G N1,N2 . Since the group G o N1,N2 decomposes as G p1q N1ˆG p2q N2 , we can also write τ π " τ p1q π b τ p2q π , with τ piq π an irreducible cuspidal representation of G piq Ni . The irreducible cuspidal representations τ of the groups G o N1,N2 were classified by Lusztig [32,33], in terms of semisimple elements s of the dual group and unipotent irreducible cuspidal representations of the centralizer of s, generalizing the classification of Green. (The only unipotent irreducible cuspidal representation of GL n pk F q is the trivial representation of GL 1 pk F q.) We will recall this later, in Section 7, when we require it.

Reducibility of parabolic induction
In this section, we recall some basic results, in particular due to Silberger, on reducibility of parabolic induction. We continue with the same notation, so that G " UpVq o is our classical group. We recall that we have the group G n " UpV n q o , with Levi subgroup M n » GL n pFqˆG (with the isomorphism determined by the chosen Witt basis) and standard parabolic subgroup P n " M n N n . Let ρ P A n pFq and π P ApGq, so that we can consider ρ b π as a representation of M n .
We are interested in the (ir)reducibility of the normalized parabolically induced representation Ipρ, π, sq " Ind Gn Pn ρ| detp¨q| s F b π, for s P C, where |¨| F is the normalized absolute value on F (with image q Z ) and det is the determinant on GL n pFq. We note that replacing ρ by an unramified twist just has the effect of translating the parameter s; that is Ipρ| detp¨q| s0 F , π, sq " Ipρ, π, s`s 0 q. Thus we lose no information if we replace our base-point ρ with any unramified twist.  (i) If Ipρ, π, sq is reducible for some s P R, then there exists s 0 P R such that ρ| detp¨q| s0 F is self-dual. (ii) If ρ is self-dual and Ipρ, π, sq is reducible for some s P R, then there is a (unique) real number s π pρq ě 0 such that, for s P R, Ipρ, π, sq is reducible if and only if s "˘s π pρq.
Remark 4.2. In the situation of Theorem 4.1, it is almost true that, if ρ is self-dual then Ipρ, π, sq is reducible for some s P R. The only exception comes from even special orthogonal groups, where we have an extra subtlety (see [23] for more details): if π is an irreducible cuspidal representation of G which is not normalized by the full orthogonal group G`and n " 1 (so that ρ is a trivial or quadratic character of Fˆ) then Ipρ, π, sq is irreducible for all s P R. On the other hand, in this situation, putting π`" Ind GG π, which is an irreducible cuspidal representation of G`, then Ipρ, π`, sq does have reducibility, at s " 0.
From Theorem 4.1, for a fixed π P ApGq, we get a map where we define s π pρq " 0 if Ipρ, π, sq is irreducible for all s P R. Part of the well-known "Basic Assumption" made in [42] is that the image of this map is in fact in 1 2 Z (indeed, this is now known in many cases -at least when G is quasi-split -through the work of Arthur, Moeglin, Waldspurger). We will prove here (independently) that this is indeed the case for depth zero representations π.
Silberger's results in fact give a little more than stated here, since we have stated them only for real values of s. Indeed, if ρ P A σ pFq then there are (up to equivalence) exactly two unramified twists of ρ which are self-dual: ρ and another one ρ 1 . If, moreover, ρ is a depth zero representation then this second representation is easy to describe: it is ρ 1 :" ρ| detp¨q| πi{n log q F . Thus Silberger's result in fact gives a qualitative description of all complex s for which Ipρ, π, sq is reducible.
In general, we will here only be able to compute the pair of numbers ts π pρq, s π pρ 1 qu, rather than distinguishing them individually. However, this is sufficient to prove the equality (1.3).

Covers and Hecke algebras
The theory of types and covers was developed by Bushnell-Kutzko to give a strategy and framework to describe the structure of the category of smooth representations of a connected reductive group. Here we are interested only in a rather special case (in particular, we have only maximal proper Levi subgroups and depth zero representations for classical groups) so we do not give definitions and results in their full generality. In particular, we are specializing to depth zero the results of [4, §3.2].
We continue in the notation of the previous section but restrict to depth zero. Thus we have ρ P A σ r0s pFq, a representation of GL n pFq, and π P A r0s pGq, giving us a representation ρ b π of the Levi subgroup M n » GL n pFqˆG of G n . We write ρ " c-Ind GLnpFq Jρ Λ ρ and π " c-Ind G Jπ λ π , as in Section 3, and use all the associated notation from there. We write P n " M n N n and denote by Pḿ " M n Nń the opposite parabolic subgroup.
We put J M " J ρˆJπ , a compact open subgroup of M n , and λ M " λ ρ b λ π , an irreducible representation of J M . From [36, Theorem 1.1], there is a cover pJ, λq of pJ M , λ M q, that is ‚ J is a compact open subgroup of G n which has an Iwahori decomposition with respect to pM n , P n q and such that J X M n " J M ; ‚ λ is an irreducible representation of J whose restriction to J M is λ M and whose restriction to J X Nn is a multiple of the trivial representation; ‚ the Hecke algebra H pG n , λq contains an invertible element whose support is the pJ, Jq-double coset of a strongly positive element of the centre of M n . Moreover, we have a description of the Hecke algebra H pG n , λq given by [36, Theorem 1.2]: (i) If there is some g P G n zM n which normalizes M n and such that the conjugate by g of ρ b π is equivalent to an unramified twist of ρ b π, then H pG n , λq is a generic Hecke algebra on an infinite dihedral group; that is, it is generated by T 1 , T 2 , each supported on a single pJ, Jq-double coset, with relations pT i´q fi qpT i`1 q " 0, for some half-integers f i ě 0. Moreover, there is a recipe to compute the f i , which we revisit in Section 6 below. (ii) Otherwise, H pG n , λq is abelian, isomorphic to C " Z˘1 ‰ . In the second case, the induced representation Ipρ, π, sq is irreducible for any s P C so we restrict our interest to the first case. Since ρ is self-dual, the condition in (i) is always satisfied, unless G is an even-dimensional special orthogonal group, the representation π is not normalized by the full orthogonal group G`, and n " 1. (See Remark 4.2.) We write R rρ,πs pM n q for the full subcategory of (smooth complex) representations of M n all of whose irreducible subquotients are unramified twists of ρb π, and R rρ,πs pG n q for the full subcategory of representations of G n all of whose irreducible subquotients have supercuspidal support an unramified twist of ρ b π. Then, since pJ, λq is a cover of pJ M , λ M q which is a type, we have a normalized embedding of Hecke algebras t : H pM n , λ M q ãÑ H pG n , λq giving us a commutative diagram Here, the functor t˚maps a right H pM n , λ M q-module X to Hom H pMn,λMq pH pG n , λq, Xq, where the H pM n , λ M q-module structure on H pG n , λq is given by t. The horizontal arrows are equivalences of categories: the functor h M is given by ξ Þ Ñ Hom JM pλ M , ξq, and similarly for h.
The Hecke algebra H pM n , λ M q is isomorphic to CrZ˘1s, where Z is supported on ζJ M and ζ is the element of the centre of M n which acts on V n " nH´' V ' nH`as 1 on V and as ̟ F on nH`. The element T 2 T 1 P H pG n , λq is supported on the double coset JζJ and, since tpZq is supported on the same double coset, we may (and do) normalize Z so that tpZq " T 2 T 1 . Now, from [4, Proposition 3.12], we get that, if Ipρ, π, sq is reducible, then the real part of s belongs to the set We will see these values are always half-integers. Thus, in the notation of Section 4, we have ts π pρq, s π pρ 1 qu " where |¨| is the usual (real) absolute value, and we recall that ρ 1 is the unique self-dual unramified twist of ρ which is not equivalent to ρ.

Reduction to the finite case
We now describe how to relate the parameters f i of the previous section to a problem in the representation theory of finite reductive groups, and rephrase the equality (1.3) in these terms. The recipe for computing these parameters is described in [36, §6.3], which is considerably simplified by only treating depth zero representations; in particular, the character χ of loc. cit. is trivial. Thus the content of this section is all proved in loc. cit. and here we just explicate it in our special case.
We recall that π " c-Ind G Jπ λ π , where J π is the normalizer of a standard maximal parahoric subgroup J o N1,N2 , with N 1 , N 2 uniquely determined, and N2 . More explicitly, as in Section 2, we write for i " 1, 2.
We will also write G piqǸ i for the full finite classical group of which G piq Ni is the connected component. We will need to distinguish the cases when τ piq π is normalized by G piqǸ i from those where it is not.
By construction, the group J in G n has reductive quotient isomorphic to GL n pk F qĜ N1,N2 , and we denote by J o the inverse image of its connected component. The parahoric subgroup J o is contained in precisely two maximal compact subgroups of G n , namely the standard maximal compact J 2 :" J N1,N2`n and a maximal compact group J 1 conjugate to J N1`n,N2 . More precisely, both J 1 , J 2 would be standard with respect to the ordered Witt basis eŃ , . . . , eŃ 1`1 , eŃ`n, . . . eŃ`1, eŃ 1 , . . . , e1 , e an 1 , . . . , e an N an , e1 , . . . , eǸ 1 , eǸ`1, . . . , eǸ`n, eǸ 1`1 , . . . , eǸ . These maximal compact subgroups have reductive quotients G 1 , G 2 isomorphic to G N1`n,N2 and G N1,N2`n respectively, and the image of J o in each of these is a parabolic subgroup with Levi component isomorphic to GL n pk F qˆG o N1,N2 . More explicitly, the G i have connected components Moreover, we have certain Weyl group elements s i P J i , defined in [36, §5.6]. (See also [51, §6.2] where, in most cases, they are denoted s ̟ 1 , s 1 respectively, though there is an added complication when n " 1 and G is a special orthogonal group, explained in [36, §5.6].) More explicitly, both s 1 , s 2 exchange (up to scalars) the vectors eǸ`j and eŃ`j, for 1 ď j ď n and preserve the subspace V of V n . Now let pJ, λq be the cover of pJ ρˆJπ , λ ρ b, λ π q from Section 5. The proof of the existence of this cover, in [36], goes via first constructing a cover pJ o , λ o q of pJ ρˆJ o π , λ ρ b λ o π q. The Hecke algebras H pG n , λ o q and H pG n , λq are not in general isomorphic, but they are closely related, as we now describe.
The Hecke algebra H pG n , λq is generated by two elements T 1 , T 2 , with T i supported on Js i J and satisfying a quadratic relation pT i´q fi qpT i`1 q " 0. Then: (i) If n " 1, and either G piqǸ i is the trivial orthogonal group (i.e. the orthogonal group on a trivial space) or the irreducible cuspidal representation τ Note that our assumption that we have reducibility implies that this can be the case for at most one value of i, and this is never the case if G is either symplectic or unramified unitary. (ii) Otherwise, there is a corresponding element T o i P H pG n , λ o q with support J o s i J o and satisfying the same quadratic relation as T i (with the same parameter f i ).
It remains to describe the parameter f i in the latter case and we assume from now on that we are in that situation.
By inflation, we have a support-preserving algebra injection and the algebra on the right is described (at least in the case that the ambient finite classical group has connected centre) in [35,Theorem 8.6]: they are two-dimensional, generated by an element satisfying the same quadratic relation, with q fi the quotient of the dimensions of the two irreducible factors of the representation parabolically induced from τ ρ b τ piq π . Moreover, one can compute this by using Lusztig's Jordan decomposition of characters, as we describe in the next section.

Computation of parameters
In this section, we undertake the computation of the parameters in the finite Hecke algebras from above. When the finite reductive group arising has connected centre, this can more-or-less be read off from the Jordan decomposition of characters and the case of unipotent irreducible cuspidals, for which there are tables in [34]. In general, one must first embed the group into one with connected centre, and then make the comparison. A special case is already carried out in [31], where they look at the Hecke algebra coming from inducing a self-dual irreducible cuspidal representation of the Siegel parabolic of a classical group.
In order to fit with the usual notations for finite reductive groups, the notation in this section is independent of that in the rest of the paper. We will not recall here the definitions of geometric and rational Lusztig series, both of which give partitions of the set of irreducible representations of a connected finite reductive group coming from Deligne-Lusztig induction; we refer the reader instead, for example, to [13] or [15]. 7.1. Self-dual polynomials. We begin with a brief section on irreducible self-dual polynomials over finite fields, since these will be used to parametrize the irreducible cuspidal representations of our finite reductive groups. We fix F q a finite field of odd cardinality q and let F qo be a subfield of index at most 2. We denote by σ the automorphism generating GalpF q {F qo q, and use the same notation for the induced automorphism of the polynomial ring F q rXs, obtained by applying σ to all coefficients.
We say that a monic polynomial P P F q rXs is F q {F qo -self-dual if P " P σ ; thus P is F q {F qo -self-dual if and only if: (i) when F q " F qo , for each root ζ of P (in some algebraic closure of F q ), the element ζ´1 is also a root of P ; (ii) when F q ‰ F qo , for each root ζ of P , the element ζ´q o is also a root of P . When F q " F qo , we will just speak of self-dual polynomials; these might more often elsewhere be called reciprocal.
If we now restrict to irreducible F q {F qo -self-dual monic polynomials P , the possibilities are somewhat constrained: (i) when F q " F qo , either P pXq " X˘1 or else degpP q is even; (ii) when F q ‰ F qo , we must have that degpP q is odd.
7.2. Connected centre. We now turn to the problem at hand, beginning in the case of a group with connected centre, so that the centralizer of any semisimple element of the dual group is connected. Let G be a connected reductive group of classical type, over a finite field F q of odd characteristic p, with connected centre and with Frobenius map F . By classical type here we mean that G is one of: (i) an odd-dimensional special orthogonal group SO 2N`1 ; (ii) a group of symplectic similitudes GSp 2N ; (iii) a group of orthogonal similitudes GSO2 N or GSO2 N , of Witt index N , N´1 respectively. (Note that we mean here that GSO2 N is the connected component of the full group of orthogonal similitudes GO2 N .) In this case we do not allow the group GSO2 .
In each case, the Frobenius map F is the standard one. We denote by G˚the dual group and write F again for the (dual) Frobenius on it. The dual group acts naturally on an F q -vector space V, with an F q -structure and a form, of dimension 2N, 2N`1, 2N respectively in the three cases above. In case (iii), we say that V is of type`1 if it has Witt index N , and of type´1 otherwise; we say that the zero space has type`1.
Write E pG F q for the set of equivalence classes of irreducible (complex) representations of G F . Then (see for example [32, §7.6]) there is a partition into geometric Lusztig series where s runs over the conjugacy classes of semisimple elements of G˚, F . (Note that rational and geometric conjugacy classes coincide as the centre of G is connected.) The partition is given as follows: for any F -stable maximal torus T of G˚containing s, we have the Deligne-Lusztig representation R G T s; then an irreducible representation π of G F lies in E pG F , sq if and only if there is such a torus T with xπ, R G T sy ‰ 0. (Here, x¨,¨y denotes the natural G-invariant inner product on class functions, and we identify (equivalence classes of) representations with their characters.) Given a semisimple element s P G˚, F , the centralizer Gs is a connected reductive group of the same rank as G, though in general it is not a Levi subgroup. Then the Jordan decomposition of characters [32, Corollary 7.10] (see also [13,Theorem 15.8 with the following properties (see [32, §7.8]): ‚ for any irreducible representation π in E pG F , sq and any F -stable maximal torus T containing s, ‚ if the identity components of the centres of G˚and Gs have the same F q -rank then π in E pG F , sq is cuspidal if and only if ψ G s pπq is cuspidal; otherwise, no representation in E pG F , sq is cuspidal.
(We remark that all the results so far extend to the case of disconnected centre once we replace the geometric Lusztig series with the rational series -see below in section 7.3 for the notion of rational Lusztig series, which coincides with the geometric one for groups with connected cntre.) Thus we get a classification of the irreducible cuspidal representations of G F from a classification of pairs ps, τ q, with s a semisimple element of G˚, F (up to conjugacy) such that the identity components of the centres of G˚and Gs have the same F q -rank, and τ an irreducible cuspidal unipotent representation of G˚, F s (up to equivalence). Lusztig classified the irreducible cuspidal unipotent representations of classical groups -in particular, there is at most one irreducible cuspidal unipotent representation for each such group -and we find (see [32, p172]) the following.
For s P G˚, F semisimple, we denote by P s P F q rXs its characteristic polynomial (as an automorphism of the space V on which G˚acts naturally), by V`the p`1q-eigenspace and by V´the p´1q-eigenspace. Then there is a bijection between the (equivalence classes of) irreducible cuspidal representations of G F and the set of conjugacy classes of semisimple elements s P G˚, F such that P s pXq " ź P P pXq aP , where the product runs over all irreducible self-dual monic polynomials over F q and the integers a p satisfy: ‚ for P pXq ‰ pX˘1q, we have a P " 1 2 pm 2 P`mP q, for some integer m P ě 0; ‚ writing a`:" a pX´1q and a´:" a pX`1q , there are integers m`, m´ě 0 such that (i) if G " SO 2N`1 then a`" 2pm 2`m`q and a´" 2pm 2`m´q , (ii) if G " GSp 2N then a`" 2pm 2`m`q`1 and a´" 2m 2 , (iii) if G " GSO2 N then a`" 2m 2 and a´" 2m 2 , where, in case (iii), V˘is an even-dimensional orthogonal space of type p´1q m˘, and the same in case (ii) for V´only.
Let s be a semisimple element of G˚, F and suppose that we have an F -stable Levi subgroup L˚contained in an F -stable parabolic subgroup P˚of G˚such that s P L˚. Correspondingly, we have an F -stable Levi subgroup L contained in an F -stable parabolic subgroup P of G. Then Ls is an F -stable rational Levi subgroup of Gs (though it need not be a proper Levi subgroup), contained in an F -stable parabolic Ps . Then we have a diagram The vertical arrows here are parabolic induction -i.e. Harish-Chandra induction -and we have abbreviated from Ind G F L F ,P F since the notation is already heavy.) This diagram commutes (this is a result of Shoji, which can be extracted from the appendix to [16]); in the cases that interest us here it can be seen fairly directly: ‚ If Gs Ď L˚then the vertical arrows preserve irreducibility (this is a special case of [32, (7.9.1)]) and the diagram commutes by (7.1). In fact, this also generalizes to the case where the parabolic P is not F -stable, replacing Ind G L,P by the Deligne-Lusztig map ε G ε L R G LĂP ; indeed, in that case the diagram commutes by definition of the map ψ G s (see the proof of [32, Proposition 7.9]).
‚ Suppose L is a maximal proper Levi subgroup and τ P E pL, sq is cuspidal. Then N G pLq{L has order 1 or 2. We are interested in the case where Ind G L τ is reducible; equivalently, N G pLq{L has order 2 and, writing w for a representative of the nontrivial coset, w normalizes τ . In this case the induced representation decomposes as (the inequality is strict by [35,Theorem 8.6]) and End G F pInd G L,P τ q is a twodimensional algebra with a quadratic generator T satisfying a relation of the form Moreover, the same is true for Ind Gs Ls ,Ps ψ L s pτ q and the recipe given in [35, §8] (see op. cit. Theorem 8.6) to calculate f τ depends on the computation of certain Weyl groups which are the same for both induced representations. (Indeed, this matching is the idea behind the inductive proof of the Jordan decomposition of characters.) On the side of the centralizer, we have unipotent representations and the parameter q fτ can be read from the tables in [34]. In the special case that s 2 " 1 (to which one could reduce) one can also read off the parameter from [35, Proposition 8.3].
7.3. Disconnected centre. We now consider the case which really interests us here. So we suppose that G is a classical group over a finite field F q of odd characteristic p, with Frobenius map F . By classical group here, we mean that G is one of: (i) an odd-dimensional special orthogonal group SO 2N`1 ; (ii) a symplectic group Sp 2N ; (iii) an even-dimensional special orthogonal group SO2 N or SO2 N , of Witt index N , N1 respectively, where again we do not allow the group SO2 . In each case, the Frobenius map F is again the standard one. Case (i) has already been treated above, so we only consider cases (ii),(iii) here.
In each case, we embed G in a group r G with connected centre of the type considered in Section 7.2. Then we get a map r G˚Ñ G˚which maps conjugacy classes of semisimple elements in r G˚, F to (rational) G˚, F -conjugacy classes of semisimple elements in G˚, F . The geometric conjugacy class of a semisimple element s in G˚, F splits into two G˚, Fconjugacy classes if and only if its centralizer Gs is disconnected, which happens if and only both 1 and´1 are eigenvalues of s. Here we also have a partition (into rational Lusztig series) of the set of equivalence classes of irreducible representations of G F , where s runs over the G˚, F -conjugacy classes of semisimple elements of G˚, F . Each geometric Lusztig series is the union of at most two rational Lusztig series, corresponding to the rational conjugacy classes in a geometric conjugacy class. Moreover, ifs P r G˚, F maps to s P G˚, F , then the rational series E pG F , sq is precisely the set of irreducible components of the restriction to G F of the representations in the Lusztig series E p r G F ,sq (see [13,Proposition 15.6]).
We begin by considering the irreducible cuspidal representations of G F . An irreducible representation of G F is cuspidal if and only if it is a component of the restriction of an irreducible cuspidal representation of r G F . In general, an irreducible cuspidal representation of r G F will decompose as a sum of at most two pieces on restriction to G F , inequivalent but of the same dimension when there are two (since they are conjugate by r G F ). Precisely what happens is essentially determined by [32,Lemma 8.9], which treats the special case of quadratic unipotent representations, as follows.
Lets P r G˚, F be such that its image s P G˚, F is an involution and such that E p r G F ,sq contains an irreducible cuspidal representation π; then π| G F is irreducible if and only if s "˘1. (Note that s "´1 is in fact not possible when G " Sp 2n .) This is enough to deal with the general case because of the following (see [13,Theorem 8.27]). Suppose L is an F -stable Levi subgroup of G and P is a parabolic subgroup (not necessarily F -stable) with Levi L, and let L˚, P˚be the corresponding subgroups of G. Suppose s P G˚, F is a semisimple element such that G˚, o Finally, this map respects cuspidality when the connected centres of G˚and L˚have the same F q -rank: i.e. in that case, π P E pL F , sq is cuspidal if and only if ε G ε L R G LĂP π in E pG F , sq is cuspidal. Now, given a semisimple elements of r G˚, F , mapping to s P G˚, F , such that E p r G F ,sq contains a cuspidal representation, denote by V " À P V P ' V 0 the decomposition of V into the rational eigenspaces corresponding to the irreducible monic self-dual polynomials of even degree, with V 0 " V`' V´the p˘1q-eigenspace; then the stabilizer L˚of this decomposition is an F -stable Levi subgroup containing the centralizer of s, and the connected centres of G˚and L˚have the same F q -rank. Thus we have a bijection between the irreducible cuspidal representations in E pG F , sq and in E pL F , sq. Moreover, denoting by r L, r P the inverse images of L, P in r G respectively, we have (see [13, (15.5)]) a commutative diagram where we have omitted the superscripts F in the functors, for ease of notation.
Putting this together, we get that, for general semisimples of r G˚, F , mapping to s P G˚, F , and π an irreducible cuspidal representation in E p r G F ,sq, the restriction π| G F is irreducible if and only if s acts as˘1 on V 0 . Thus, in fact, the restriction remains irreducible if and only if the centralizer Gs is connected. We have proved: Lemma 7.3. Let π P E pG F , sq be an irreducible cuspidal representation and lets P r G˚, F be a semisimple element mapping to s. The following are equivalent: (i) π extends to an irreducible representation in E p r G F ,sq; (ii) the centralizer Gs is connected; (iii) at most one of˘1 is an eigenvalue of s. Now we turn to the computation of the parameter. Thus, as in Section 7.2, we have s a semisimple element of G˚, F and we suppose that L˚is a maximal proper F -stable Levi subgroup contained in an F -stable parabolic subgroup P˚of G˚such that Gs Ď L˚. Correspondingly, we have F -stable Levi and parabolic subgroups L, P in G. We lift s tos P r G˚and likewise have lifts r L˚and r P˚, with r Gs Ď r L˚, and corresponding subgroups r L, r P in r G into which L, P respectively embed. We are given a cuspidal representation τ in E pL F , sq, so that τ appears (with multiplicity one) in the restriction of a cuspidal representationτ in E pL F ,sq, withs P r L˚, F mapping to s. Let w be any representative for the non-trivial element of N G pLq{L. We will eventually be interested in the case where w normalizes τ (so that the parabolically induced representation from τ is reducible) but, in that case, it is not immediately clear whether or not w normalizesτ . We will need to know exactly when this is the case but we can already say something about the parameters in the Hecke algebras. For ease of notation, we again omit the superscripts F in the following.

Lemma 7.4.
(i) Suppose w normalizesτ . Then w also normalizes τ and we have an isomorphism of Hecke algebras End G pInd G L,P τ q » End r G pInd r G r L, r Pτ q. (ii) Suppose w does not normalizeτ but does normalize τ . Then End G pInd G L,P τ q has generator T satisfying pT`1qpT´1q " 0.
Proof. We write If ℓ " 2 then we order the terms so that dimpπ 1 q ě dimpπ 2 q, and End G pInd G L,P τ q is a two-dimensional algebra with a quadratic generator T satisfying a relation of the form pT`1qpT´q fτ q " 0, q fτ " dimpπ 1 q dimpπ 2 q .
In either case, from Mackey we also have Since the dimensions are distinct, comparing with (7.5), we must have ℓ " 2 and in particular we get dimpπ 1 q{ dimpπ 2 q " dimpπ 1 q{ dimpπ 2 q so that the relations of the quadratic generators in End G pInd G L,P τ q and End r G pInd r G r L, r Pτ q are the same. (ii) Suppose now that w does not normalizeτ but does normalize τ (so that ℓ " 2). Then Ind r G r L, r Pτ is irreducible so the two pieces of Ind G L,P τ are conjugate under r G F so of the same dimension; in particular we get f τ " 0, as required.
It remains now to compute when the representationτ in E pL F ,sq containing τ is normalized by w. Ultimately, the answer depends on the family of groups in question, but we can already make a preliminary reduction. We assume from now on that w does indeed normalize τ .
We write L F » GL F nˆG F 0 , where G 0 is a (possibly trivial) classical group in the same family as G, and τ " τ 1 b τ 0 , so that τ 1 is a self-dual irreducible cuspidal representation of GL F n . We write r G 0 for the similitude group into which G 0 embeds; if G 0 is a trivial group (i.e. acts on a trivial space) then r G 0 is the multiplicative group. We also have an isomorphism r L F » GL F nˆr G F 0 , which can be seen as follows. We choose a Witt basis eŃ , . . . , e1 , e1 , . . . , eǸ for the space on which G F acts, with respect to which L F is standard (so it is the stabilizer of the subspace xeŃ , . . . , eŃ´n`1y), where we have changed from our usual notation in the case of a non-split special orthogonal group, with e1 , e1 a basis for the anisotropic part of the space. We write µ : r G F 0 Ñ Fq for the similitude map, which is the identity map when G 0 is trivial. Then we get an isomorphism GL F nˆr G F 0 Ñ r L from the map pg, hq Þ Ñ diagpg, h, µphqw n pg´1q T w´1 n q, for g P GL F n , h P r G F 0 , where w n is the antidiagonal element of GL F n with all non-zero entries equal to 1 (a representative for the longest element of the Weyl group), g T denotes the transpose matrix, and the matrix on the right hand side is block diagonal. (Note that, when G 0 is trivial, the central term h in the block diagonal matrix is acting on a trivial space, so is not really present.) Thus we can writeτ " τ 1 bτ 0 , whereτ 0 is an irreducible cuspidal representation of r G F 0 whose restriction to G F 0 contains τ 0 . We denote by ε the sign such that G preserves an ε-symmetric form. Then, if either G is a symplectic group or n is even, we can take the representative w given by thus w is block antidiagonal (for the block sizes corresponding to L), and the conjugation action of w on r L, transported back to GL F nˆr G F 0 , is given by (7.6) pg, hq Þ Ñ pµphqpg´1q T , hq.
Since τ 1 is self-dual, if its central character is also trivial then the map (7.6) clearly intertwines τ 1 bτ 0 with itself. The only case when the central character is non-trivial is when τ 1 is the quadratic character, where we see that (7.6) intertwines τ 1 bτ 0 with itself if and only ifτ 0 »τ 0 b pτ 1˝µ q. This leaves the case of even special orthogonal groups with n " 1, where we need a different representative w. Note that, in this case, we do not have that G 0 is the trivial group, since we have excluded the case G " SO2 . We have an identification G 0 » SO2 pN´1q , from the action on xeŃ´1, . . . , eǸ´1y, and we pick c 0 P O2 pN´1q z SO2 pN´1q . Then we can take w to be the element given by # wpeN q " eN , w| xeŃ´1,...,eǸ´1y " c 0 .
The action of w on r L is given by pg, hq Þ Ñ pµphqg, c 0 hc´1 0 q and we see thatτ is normalized by w if and only ifτ 0 »τ c0 0 b pτ 1˝µ q.
If we set c 0 " 1 in the case of symplectic groups, we can unify the discussion above into the following statement.
(i) If n ą 1 thenτ is normalized by w. (ii) If n " 1 thenτ is normalized by w if and only ifτ 0 »τ c0 0 b pτ 1˝µ q. In the following subsections, we analyze precisely when the conditions in Lemma 7.7(ii) are satisfied, in terms of the eigenvalues of the semisimple element s such that τ P E pL F , sq.

Symplectic groups.
We begin with the case of symplectic groups, so that we are in case (ii) of Section 7.3. Thus we have a cuspidal representation τ in E pL F , sq, for some maximal proper Levi subgroup L of G, and a cuspidal representationτ in E p r L F ,sq whose restriction to L contains τ . We denote by w a representative for the non-trivial element of N G pLq{L, which we assume normalizes τ .
We write L F » GL F nˆG F 0 , where G 0 is a (possibly trivial) symplectic group, and τ " τ 1 b τ 0 , so that τ 1 is a self-dual irreducible cuspidal representation of GL F n . Then τ 1 P E pGL F n , s 1 q and τ 0 P E pG F 0 , s 0 q, for some semisimple elements s 0 , s 1 of the respective dual groups. Note that, if G 0 is the trivial symplectic group then its dual group is SO 1 so that s 0 " 1.
Lemma 7.8. The representationτ is normalized by w unless τ 1 is the non-trivial quadratic character of GL F 1 and´1 is not an eigenvalue of s 0 . We remark that, since 1 is always an eigenvalue of s 0 in the case of symplectic groups, the condition that´1 not be an eigenvalue of s 0 is equivalent to the condition that τ 0 extend to the similitude group r G F 0 , by Lemma 7.3. Proof. By Lemma 7.7, we only need to consider the case that τ 1 is the non-trivial quadratic character; in that case, w normalizesτ if and only ifτ 0 »τ 0 b χ, where we recall thatτ 0 is an irreducible cuspidal representation of the similitude group r G F 0 " GSp F 2pN´1q containing τ 0 , and χ " τ 1˝µ is the non-trivial quadratic character of r G F 0 . Denote by Z p r G F 0 q the centre of r G F 0 ; thenτ 0 b χ »τ 0 if and only if the restriction ofτ 0 to the index two subgroup Z p r G F 0 qG F 0 is reducible, i.e.τ 0 restricts reducibly to G F 0 . Thusτ w »τ if and only if τ 0 does not extend toG F 0 , and we are done. 7.5. Even special orthogonal groups. Now we turn to the case of even-dimensional orthogonal groups, so that we are in case (iii) of Section 7.3. Here, as well as proving the analogue of Lemma 7.8, we need to consider the cases, seen in Section 6, where the parameter in the affine Hecke algebra is zero, rather than matching the parameter in the finite Hecke algebra for the connected reductive quotient. This happens precisely when either we have a "trivial orthogonal group" or an irreducible cuspidal representation of an even-dimensional special orthogonal group which is not normalized by the full orthogonal group. Thus we must also identify when this happens, in the language of the previous sections. We begin with this question, since the answer is also needed for the proof of the analogue of Lemma 7.8. Proposition 7.9. Let τ P E pSO˘, F 2N , sq be an irreducible cuspidal representation, and letτ P E pGSO˘, F 2N ,sq be an irreducible cuspidal representations, wheres is a semisimple element mapping to s. (ii) τ extends to a representation of O˘, F 2N if and only if at least one of˘1 is an eigenvalue of s.
We remark that in the case that s 2 " 1 (that is, τ is a quadratic unipotent representation), (i) is proved in [32,Lemma 8.9], while (ii) is proved in [53] (see the proof of op. cit. Proposition 4.3). Our proofs in the general case are similar.
Proof. In order to prove this, we need to recall a little about the dual group of GSO2 N . This dual group is the special Clifford group C 0 pVq, which sits in an exact sequence which is exact on points by Hilbert's Theorem 90. It is the connected component of the full Clifford group CpVq, which sits in a similar exact sequence, mapping onto the full orthogonal group OpVq. The Clifford group is a subgroup of the group of invertible elements of the Clifford algebra A " ApVq, which is Z{2Z-graded A " A 0 ' A 1 ; then CpVq " C 0 pVq \ C 1 pVq, with C i pVq " CpVq X A i , and the special Clifford group is a subgroup of index two in the full Clifford group.
We will need to know when the semisimple elements P C 0 pVq of the special Clifford group is centralized by some element of C 1 pVq. If 1 is an eigenvalue of s then it has an anisotropic eigenvector v with eigenvalue 1 and one checks that v P C 1 pVq is centralized bys. If 1 is not an eigenvalue of s, then the elements of A 1 which commute withs are linear combinations of elements of the form v 1¨¨¨vr , with v i linearly independent eigenvectors of s with eigenvalue ζ i , such that ś r i"1 ζ i " 1 and r is odd. However, any two such elements anti-commute (since r is odd) and any such element squares to 0, since the v i are isotropic unless ζ i "´1 and we cannot have all ζ i "´1, since their product is 1 and r is odd. Thus any element of a P A 1 which commutes with s satisfies a 2 " 0 so is non-invertible. Thus no element of C 1 pVq commutes withs.
Finally, we pick c P O˘, F 2N z SO˘, F 2N and we are ready to begin the proof. (i) The representationτ extends to GO˘, F 2N if and only if it is normalized by some c. Now τ c P E pSO˘, F 2N ,c´1scq, for somec P C 1 pVq. On the other, hand, these Lusztig series contain only one cuspidal representation each, soτ »τ c if and only ifs is conjugate in C 0 pVq toc´1sc, that is, if and only if the centralizer in CpVq ofs is not contained in the special Clifford group C 0 pVq. However, we have seen above that this happens if and only if 1 is an eigenvalue of s. This leaves the case when both˘1 are eigenvalues of s. Letτ be an irreducible cuspidal representation in E pGSO˘, F 2N ,sq whose restriction to SO˘, F 2N contains τ . By (i), this representation is normalized by c. Now we consider the representation 1 b τ of L F " GL F 1ˆS O˘, F 2N , a Levi subgroup of SO˘, F 2pN`1q . As at the end of Section 7.3, we choose a Witt basis eŃ`1, . . . , eǸ`1 with respect to which L F is standard and denote by w the element of SO˘, F 2pN`1q given by # wpeN`1q " eN`1, w| xeŃ ,...,eǸ y " c.
Similarly, we have the representation 1bτ of r L F " GL F 1ˆG SO˘, F 2N , which is normalized by w, by Lemma 7.7(ii). But then Lemma 7.4(i) implies that w also normalizes 1 b τ , whence τ is normalized by c. Thus τ extends to O˘, F 2N , as required. Now we return to the notation at the end of section 7.3. Thus, with G " SO2 N and r G " GSO2 N , we have a cuspidal representation τ in E pL F , sq, for some maximal proper Levi subgroup L of G, and a cuspidal representationτ in E p r L F ,sq whose restriction to L contains τ . We denote by w a representative for the non-trivial element of N G pLq{L, which we assume normalizes τ .
We write L F » GL F nˆG F 0 , where G 0 is a (possibly trivial) special orthogonal group, and τ " τ 1 b τ 0 , so that τ 1 is a self-dual irreducible cuspidal representation of GL F n . Then τ 1 P E pGL F n , s 1 q and τ 0 P E pG F 0 , s 0 q, for some semisimple elements s 0 , s 1 of the respective dual groups. Note that, if G 0 is the trivial special orthogonal group then its dual group is the trivial group so that s 0 has no eigenvalues.
Corollary 7.10. The representationτ is normalized by w in all but the following cases: (i) τ 1 is the trivial character of GL F 1 and 1 is not an eigenvalue of s 0 ; (ii) τ 1 is the quadratic character of GL F 1 and´1 is not an eigenvalue of s 0 . We remark that, among the exceptional cases, we cannot have that G 0 is the trivial group, since otherwise we would have G " SO2 , which we have excluded.
Proof. By Lemma 7.7, we need only consider the case n " 1, so that τ 1 is a trivial or quadratic character; thenτ is normalized by w if and only ifτ 0 »τ c0 0 b pτ 1˝µ q, where we recall thatτ 0 is an irreducible cuspidal representation of the similitude group r G F 0 " GSO˘, F 2pN´1q containing τ 0 , and c 0 P O˘, F 2pN´1q z SO˘, F 2pN´1q . Suppose first that τ 1 is trivial. Thenτ is normalized by w if and only ifτ 0 is normalized by c 0 , which happens if and only ifτ 0 extends to the full similitude group; by Proposition 7.9, this happens if and only if 1 is an eigenvalue of s 0 , and we are done. Now suppose that τ 1 is the non-trivial quadratic character and put χ 1 " τ 1˝µ . We also denote by c 1 an element of GSO˘, F 2pN´1q which is not in Z pGSO˘, F 2pN´1q q SO˘, F 2pN´1q ; thusτ 0 is an extension of τ 0 if and only if τ 0 is normalized by c 1 , if and only ifτ 0 »τ 0 bχ 1 .
If both˘1 are eigenvalues of s 0 , thenτ 0 »τ b χ 1 , whileτ 0 extends to GO˘, F 2pN´1q , by Proposition 7.9(i). Thusτ c0 0 b χ 1 »τ 0 b χ 1 »τ 0 . If neither˘1 is an eigenvalue of s 0 , thenτ c0 0 b χ 1 is an extension of τ c0 0 , which is not equivalent to τ 0 by Proposition 7.9(ii). Thusτ c0 0 b χ 1 fiτ 0 . Finally, if exactly one of˘1 is an eigenvalue of s 0 , then τ c0 0 » τ 0 , by Proposition 7.9(ii). Thenτ c0 0 contains τ 0 on restriction, so is equivalent either toτ 0 or toτ 0 b χ 1 , since it agrees withτ 0 on the index two subgroup Z pGSO˘, F 2pN´1q q SO˘, F 2pN´1q of GSO˘, F 2pN´1q . By Proposition 7.9(i), the former happens if and only if 1 is an eigenvalue of s 0 , in which caseτ c0 0 bχ 1 »τ 0 bχ 1 fiτ 0 . Thus the latter happens if and only if´1 is an eigenvalue s 0 , in which caseτ c0 0 b χ 1 »τ 0 . 7.6. Summary. We summarize the results of all these calculations, including looking up parameters in Lusztig's tables in [34], in the following table. We are given an irreducible cuspidal representation τ in E pL F , sq, for some maximal proper Levi subgroup L of G, which is normalized by N G pLq. We write L F » GL F nˆG F 0 , where G 0 is a (possibly trivial) classical group of the same type as G, and τ " τ 1 b τ 0 , so that τ 1 is a self-dual irreducible cuspidal representation of GL F n . Then τ 1 P E pGL F n , s 1 q and τ 0 P E pG F 0 , s 0 q, for some semisimple elements s 0 , s 1 of the respective dual groups.
We write P s0 pXq " ź P P pXq aP pX´1q a`p X`1q af or the characteristic polynomial of s 0 , where the product is over all irreducible selfdual monic polynomials over F q of even degree, and the integers a P , a˘are related to integers m P , m˘as in the description in (7.2). We also write Q for the characteristic polynomial of s 1 P GL˚, F n ; thus either QpXq " pX˘1q or Q is an irreducible self-dual monic polynomial of even degree n " n Q . In the table, the cases (i)-(iii) refer to the different possible classical groups, as in Section 7.3.
Unitary groups. Finally, we consider the case of unitary groups. We could have included this in the cases of Sections 7.2-7.6 above but it would have further complicated the notation. Instead, we indicate here the differences with the previous cases and summarize the final results. Let q " q 2 o be an even power of an odd prime p, take G " GL n over the finite field F qo , and let F be the twisted Frobenius map, so that G F is a unitary group (which we can think of as a subgroup of GL n pF q q). Then G˚" GL n act naturally on an n-dimensional vector space V with an F q {F qo -hermitian form. For s P G˚, F semisimple, we denote by P s pXq P F q rXs its characteristic polynomial as an automorphism of V.
From [32, §9], the equivalence classes of irreducible cuspidal representations of G F are in bijection with the set of conjugacy classes of semisimple elements s in G˚, F whose characteristic polynomial is of the form where the product runs over all irreducible F q {F qo -self-dual monic polynomials in F q rXs (see Section 7.1), and a P " 1 2 pm 2 P`mP q, for some integer m P ě 0. Now suppose L is a maximal proper F -stable Levi subgroup of G contained in an Fstable parabolic subgroup P. We write L F » GL m pF q qˆG F 0 , with G F 0 again a unitary group. Let τ be an irreducible cuspidal representation of L F with the property that any representative w for the non-trivial element of N G pLq{L normalizes τ . Thus we may decompose τ " τ 1 bτ 0 , with τ 1 a (conjugate)-self-dual irreducible cuspidal representation of GL m pF q q and τ 0 an irreducible cuspidal representation of G F 0 .
In this situation, the induced representation Ind G L,P τ decomposes again as π 1 ' π 2 , with dimpπ 1 q ą dimpπ 2 q, and End G F pInd G L,P τ q is a two-dimensional algebra with a quadratic generator T satisfying a relation of the form pT`1qpT´q fτ q " 0, q fτ " dimpπ 1 q dimpπ 2 q ą 1.
As in the connected case above, the parameter may be computed via the Jordan decomposition of characters and Lusztig's tables, as follows. For s 0 a semisimple element of G˚, F 0 such that τ 0 P E pG F 0 , s 0 q, we write its characteristic polynomial for integers a P " 1 2 pm 2 P`mP q as above. We also write Q for the irreducible characteristic polynomial of an element s 1 P GL˚, F m such that τ 1 P E pGL F m , s 1 q; thus Q is an irreducible F q {F qo -self-dual monic polynomial, of some odd degree n " n Q . Then we get f τ " p2m Q`1 q n Q 2 .

Synthesis
In this section, we put together the previous results to verify the equality (1.3), for π a depth zero irreducible cuspidal representation of the classical group G. Recall that N p G is the dimension of the vector space on which the complex dual group p G acts naturally. Strictly speaking, here we only prove the inequality ÿ by checking that the sum over depth zero self-dual irreducible cuspidal representations already gives us N p G , that is: In many cases, the opposite inequality was already proved by Moeglin in [38]; alternatively, the techniques used here, together with the results in [36], easily show that, for ρ a positive depth self-dual irreducible cuspidal representation we have s π pρq P 0,˘1 2 ( , so that these do not contribute to the sum. (See [6], where this is carried out in a more general situation, for details.) Thus we return to the notation of Sections 2-6: we have π " c-Ind G Jπ λ π an irreducible cuspidal depth zero representation of a classical group G, with J π the normalizer of a standard maximal parahoric subgroup J o N1,N2 , and λ π | J o where the product runs over irreducible k F {k o -self-dual monic polynomials in k F rXs, and the powers a piq P satisfy the conditions of (7.2); in particular, there are integers m piq p ě 0 such that: ‚ if k F ‰ k o or P pXq ‰ pX˘1q then a piq P " 1 2 m piq P pm piq P`1 q; ‚ if k F " k o and P pXq " pX˘1q then we write m piq " m piq pX´1q and m piq " m piq pX`1q , to match the notation of Section 7, and these satisfy the conditions in (7.2).
Remark 8.2. It may be that N i " N an i " 0, for i " 1 or 2; in this case the group G piq Ni is trivial, but we must interpret it as the "right" trivial group. That is, if G is symplectic then the group is a trivial symplectic group; if G is special orthogonal it is a trivial special orthogonal group; if G is unramified unitary it is a trivial unitary group; and if G is ramified unitary then it is a trivial symplectic group if ε " p´1q i , and a trivial special orthogonal group otherwise. In particular, if the group is trivial symplectic then the characteristic polynomial of s i is X´1; in the other cases, the characteristic polynomial of s i is the constant polynomial 1. Now, for ρ a self-dual irreducible cuspidal depth zero representation of some GL n pFq, we have a unique self-dual irreducible cuspidal representation τ ρ of GL n pk F q such that ρ contains the representation λ ρ of GL n po F q obtained from τ ρ by inflation. Then τ ρ is in the Lusztig series associated to some conjugacy class in GL n pk F q with irreducible self-dual characteristic polynomial Q " Q ρ of degree n.
We suppose first that k F ‰ k o or QpXq ‰ pX˘1q; thus either n ą 1 or G is an unramified unitary group, and the parameters q fi of the Hecke algebra are always computed from the Hecke algebra in the finite group. Then the formulae in Sections 7.6-7.7 give piq Q`1 q n 2 and, from (5.1) we get reducibility points ˘s π pρq,˘s π pρ 1 q ( " Since one of these is an integer and the other a half-integer, we get ts π pρq 2 u`ts π pρ 1 q 2 u "˜p m Thus we are already done in the case of unramified unitary groups: summing, we get ÿ ρPA σ r0s pFq ts π pρq 2 un ρ " ÿ P pa p1q P`a p2q P q degpP q " p2N 1`N an 1 q`p2N 2`N an 2 q " 2N`N an , as required. For the cases k F " k o and QpXq " pX˘1q, we will split according to the type of group G, since the values for the parameters do not admit such a uniform description. 8.1. Symplectic groups. We suppose first that QpXq " X´1, so that ρ, ρ 1 are the trivial character and the unramified character of order 2, and write m piq Q " m piq . Since both G piq Ni are symplectic groups, we get with reducibility points ˘s π pρq,˘s π pρ 1 q ( " !˘p m p1q`mp2q`1 q,˘pm p1q´mp2q q ) .

Ramified unitary groups.
In this case, the groups G piq Ni are one symplectic and one orthogonal; for ease of exposition, we will assume that G p1q N1 is symplectic (otherwise exchange 1 and 2).
We begin again with the case QpXq " X´1 and write m piq in place of m piq Q . Thus we get Thus we get reducibility points ˘s π pρq,˘s π pρ 1 q ( " 2¯) , otherwise. Thus, if N an 2 " 1, we get ts π pρq 2 u`ts π pρ 1 q 2 u " pm p1q`mp2q`1 q 2`p m p1q´mp2q q 2 " a p1q`ap2q ; and otherwise, since both reducibility points are half-integers, we get ts π pρq 2 u`ts π pρ 1 q 2 u "´m The case QpXq " X`1 is similar, the main difference being that f 1 " 2m p1q . Then we get reducibility points ˘s π pρq,˘s π pρ 1 q ( " Now in both cases we get ts π pρq 2 u`ts π pρ 1 q 2 u " a p1q`ap2q .
Noting that we have we once again see that, summing over all depth zero self-dual irreducible cuspidal representations of all GL n pFq, equation 8.1 is satisfied. 8.3. Special orthogonal groups. The case of special orthogonal groups is exactly analogous and we do not give the details. One can check the equality in (8.1) by working through the cases according to the parities of N an 1 , N an 2 . For example, if both are odd, then with QpXq " X˘1 we get ts π pρq 2 u`ts π pρ 1 q 2 u " a p1q`ap2q`1 .
The additions of the extra 1 here exactly compensate for the fact that the dual groups of G We have also seen that s π pρq P 1 2 Z in all cases.

L-packets and Examples
In this final section, we examine the implications of the results here for the computation of L-packets and give some examples. Firstly, we recall some facts about the (expected) sizes of discrete series L-packets containing an irreducible cuspidal representation, and the (expected) number of cuspidal representations in them.
Let ϕ : W FˆS L 2 pCq Ñ p G¸W F be a Langlands parameter for G whose L-packet Π ϕ contains an irreducible cuspidal representation π of G. Then, as recalled in the introduction, we should have where ϕ ρ is the (irreducible) representation of the Weil group W F corresponding to ρ via the Langlands correspondence for general linear groups, and st m is the m-dimensional irreducible representation of SL 2 pCq. Putting ℓpπq " # Jordpπq, the number of representations one expects in the packet Π ϕ is 2 ℓpπq´1 , since this is the number of characters of the component group of Cent p G pImpϕqq trivial on the centre of p G. We also set Epπq " tρ P A σ pFq : s π pρq P Nu and epπq " #Epπq; this is the number of ρ P A σ pFq such that pρ, mq P Jordpπq for some odd integer m. Finally, put e 0 pπq " # 1 if there is ρ P A σ pFq such that s π pρq P 1`2N, 0 otherwise.
Then the number of irreducible cuspidal representations one expects in the L-packet Π ϕ is 2 epπq´e0pπq , since this is the number of characters of the component group of Cent p G pImpϕqq which are trivial on the centre of p G and alternating, in the sense of, for example, [40,Section 8].
Remark 9.1. The reference to the work of Moeglin in [40,Section 8] is in fact only for unitary groups, though it also holds when G is a quasi-split unitary, orthogonal, symplectic or GSpin group (see [39]) and it seems reasonable to expect it to hold in more generality. Although our results here on the representations with given inertial reducibility set do not require it, for the purposes of discussion, from now on we make the following assumption: (A) The description of the number of irreducible cuspidal representation in an Lpacket above is valid for the group G.
However, extra care must be taken when G is an even-dimensional special orthogonal group -see Example 9.6. Now we describe how our results allow us to find all irreducible cuspidal representations with the same inertial reducibility set, hence all irreducible cuspidal representations in a union of one, two or four L-packets (assuming (A)). We suppose we are in the general situation of the previous section, with π an irreducible cuspidal depth zero representation of G. Recall that IRedpπq " ttprρs, mq : ρ P A σ pFq, m P N with 2s π pρq " m`1uu , where rρs denotes the inertial equivalence class of ρ.
Our representation π is induced from a representation containing the inflation of an irreducible cuspidal representation τ π » τ p1q π b τ p2q π of the reductive quotient G p1q N1ˆG p2q N2 of a maximal parahoric subgroup. For i " 1, 2 and P an irreducible k F {k o -self-dual monic polynomial in k F rXs, we denote by m piq P the associated non-negative integer as in (7.2). The formulae obtained above show that, for each irreducible k F {k o -self-dual monic polynomial P in k F rXs, the pair of integers tm p1q P , m p2q P u can be recovered from the reducibility points ts π pρq, s π pρ 1 qu, for ρ " ρ P a representation in A σ r0s pFq with associated characteristic polynomial P , and ρ 1 its self-dual unramified twist. Indeed, one gets the following, where we write |¨| 8 for the usual (archimedean) absolute value on R: Xˇˇs π pρq˘s π pρ 1 qˇˇ8 \( ; ‚ if P pXq " X˘1 and k F " k o , so that ρ is a character of GL 1 pFq of order at most 2, then ! m p1q P , m p2q P ) " "Z |s π pρq˘s π pρ 1 q| 8

2^* .
Thus one obtains the same inertial reducibility set as for π only for irreducible cuspidal representations π 1 with tm p1q P pπq, m p2q P pπqu " tm p1q P pπ 1 q, m p2q P pπ 1 qu, for every irreducible k F {k o -self-dual monic polynomial P in k F rXs. Hence, in order to obtain other representations with the same inertial reducibility set, it is enough to exchange (some of) the integers m p1q P pπq, m p2q P pπq. Note, however, that it is not always possible to do this, for parity reasons. We set Qpπq " ! irreducible self-dual monic P P k F rXs : m p1q P pπq ‰ m p2q P pπq ) and put qpπq :" #Qpπq.
Remark 9.2. If P P k F rXs is an irreducible k F {k o -self-dual monic polynomial, with P pXq ‰ X˘1 or k F ‰ k o , and ρ P , ρ 1 P are the corresponding self-dual irreducible cuspidal representations of some GL n pFq, then one of s π pρ P q, s π pρ 1 P q is integral and the other non-integral. In particular, we see that (exactly) one of ρ P , ρ 1 P is in Epπq if and only if P P Qpπq. A similar analysis can be done for P pXq " X˘1 and k F " k o , but it depends on the type of group. We summarize the results in the separate cases below.
We will parametrize the irreducible cuspidals π 1 with IRedpπ 1 q " IRedpπq by maps ε : Qpπq Ñ t1, 2u, noting that not all such maps are permissible, and that each map may give rise to more than one representation. We split again according to the type of group. 9.1. Symplectic groups. We begin with the case that G " Sp 2N pFq is a symplectic group, in which case there are no restrictions on the maps ε : Qpπq Ñ t1, 2u. We put ) . Now suppose we have ε : Qpπq Ñ t1, 2u and, for irreducible self-dual monic P R Qpπq, we set εpP q " 1. Then we can find, for i " 1, 2, a semisimple element s piq ε in a suitable odd special orthogonal group SO 2N 1 i`1 pk F q with characteristic polynomial ź P P pXq a piq P pεq , where the product is taken over all irreducible self-dual monic polynomials in k F rXs and the integers a piq P pεq are related to integers m piq P pεq as in (7.2), with m piq P pεq " m pi¨εpP qq P , where the index is understood modulo 3. Correspondingly, we have irreducible cuspidal 1 pk F qˆSp 2N 1 2 pk F q; note that, for each ε, the number of such representations is 2 δpπq . Inflating each τ ε to the maximal parahoric subgroup J N 1 1 ,N 1 2 and inducing to G, we get an irreducible cuspidal representation. Thus we get 2 qpπq`δpπq irreducible cuspidal representations of G.
The analysis of Remark 9.2, along with that for the cases P pXq " X˘1, shows that epπq " On the other hand, we always have e 0 pπq " 1, since either s π p1q or s π pω 0 q is an odd integer, where 1 is the trivial character of GL 1 pFq and ω 0 is the unramified character of order two. Hence we have constructed the irreducible cuspidal representations in a union of two L-packets if δpπq " 2, or in a single L-packet otherwise. Thus, in some cases we are able to identify all the representations in a single L-packet, but in others we cannot distinguish between the representations in two L-packets without further work. We give some examples to illustrate these phenomena. In the following, we write ω 1 , ω 2 " ω 0 ω 1 for the (tamely) ramified characters of GL 1 pFq of order two. Example 9.3. We begin with an example where we are able to recover all the cuspidal representations in a single L-packet. We take G " Sp 6 pFq and begin with the parahoric subgroup J 2,1 , which has reductive quotient G 2,1 » Sp 4 pk F qˆSL 2 pk F q.
We take the representation θ 10 of Sp 4 pk F q, that is the unique cuspidal representation in the Lusztig series E pSp 4 pk F q, 1q (so that the associated characteristic polynomial is pX1 q 5 ). We also take an irreducible cuspidal representation τ of SL 2 pk F q in a Lusztig series with associated characteristic polynomial pX´1qpX`1q 2 . Thus τ is a representation of dimension q´1 2 , of which there are two. We denote by λ π the representation of J 2,1 inflated from θ 10 b τ and put π " c-Ind G J2,1 λ π , an irreducible cuspidal representation of G.
Following the recipe in Section 8, we find that s ρ pπq P 0,˘1 2 ( unless ρ is a character of GL 1 pFq. On the other hand, we get X`1 " 1, and hence ts π p1q, s π pω 0 qu " t2, 1u, ts π pω 1 q, s π pω 2 qu " t1u; thus IRedpπq is the multiset ttpr1s, 2q, pr1s, 1q, prω 1 s, 1q, prω 1 s, 1quu. In this case we know more since the Langlands parameter ϕ π corresponding to π has image in SO 7 pCq so, in particular, has determinant 1; thus it must be since exchanging 1 and ω 0 would give a representation with determinant ω 0 . In the notation above, we have Epπq " t1, ω 0 , ω 1 , ω 2 u so that epπq " 4. Thus the Lpacket containing π consists of 16 irreducible representations, 8 of which are cuspidal. On the other hand, we have Qpπq " tX`1, X´1u so that qpπq " 2, and δpπq " 1, so that we can construct exactly the cuspidal representations in the L-packet, as follows: (i) There is one rational conjugacy class psq in SO 3 pk F q such that its characteristic polynomial is pX´1qpX`1q 2 and the corresponding Lusztig series E pSL 2 pk F q, sq contains two irreducible cuspidal representations τ, τ 1 . Now we can inflate the representations θ 10 bτ and θ 10 bτ 1 of Sp 4 pk F qˆSL 2 pk F q to either J 2,1 or J 1,2 and then induce to G. This gives us four inequivalent irreducible cuspidal representations of G (one of which is π).
(ii) There is one rational conjugacy class ps 1 q in SO 7 pk F q such that its characteristic polynomial is pX´1q 5 pX`1q 2 and the corresponding Lusztig series E pSp 6 pk F q, s 1 q contains two irreducible cuspidal representation τ 1 , τ 1 1 . We inflate these representations to either J 3,0 or J 0,3 and induce to G, giving us another four inequivalent irreducible cuspidal representations of G, also inequivalent to those in (i).
Example 9.4. Now we look at the simplest example where the information we have so far is only sufficient to recover the cuspidal representations in a union of two L-packets. We take G " Sp 4 pFq and begin with the parahoric subgroup J 1,1 , which has reductive quotient G 1,1 » SL 2 pk F qˆSL 2 pk F q.
We take irreducible cuspidal representations τ 1 , τ 2 of SL 2 pk F q each in a Lusztig series with associated characteristic polynomial pX´1qpX`1q 2 , as in the previous example. We denote by λ π the representation of J 1,1 inflated from τ 1 b τ 2 and put π " c-Ind G J1,1 λ π , an irreducible cuspidal representation of G. Following the recipe, this time we obtain IRedpπq " ttpr1s, 1q, prω 1 s, 2quu and, using the fact that the corresponding Langlands parameter ϕ π has determinant 1, we have where ω is either ω 1 or ω 2 . However, without further work, we cannot distinguish which ramified quadratic character occurs here. This reflects the fact that m p1q X`1 " m p2q X`1 " 1 so that δpπq " 2.
Thus, at this stage, we can only identify the 4 cuspidal representations occurring in the union of the two L-packets corresponding to ω " ω 1 , ω 2 in (9.5): they are given by independently choosing the τ i to be one of the two irreducible cuspidal representations of SL 2 pk F q of dimension q´1 2 . Distinguishing these two L-packets (and identifying the two discrete series representation in each of them) requires further analysis: for this particular example, this is carried out in [5].
In general, distinguishing the representations when we have two L-packets as in Example 9.4 will probably require, as a first step, the classification of quadratic-unipotent irreducible cuspidal representations of finite classical groups, and the compatibility of this classification with Deligne-Lusztig induction, which is done by Waldspurger in [53]. 9.2. Unramified unitary groups. Suppose now that G is an unramified unitary group of dimension 2N`N an . On the one hand this case is simpler, and we will see that the set of representations with given inertial reducibility (multi)set is a single L-packet. On the other hand, we cannot arbitrarily exchange the integers m p1q P , m p2q P as above, due to parity constraints -swapping would sometimes lead to representations of the isometry group of a non-isometric hermitian space.
Recall that π is induced from the inflation of an irreducible cuspidal representa- is the reductive quotient of another maximal parahoric subgroup of G then, for i " 1, 2, the corresponding space V 1 piq must have dimension of the same parity as that of V piq .
Recall also that we have a semisimple element s i of the dual group of G piq Ni , such that τ piq π is the (unique) irreducible cuspidal representation in the corresponding Lusztig series, and that s i has characteristic polynomial ź P P pXq a piq P , where the product runs over all irreducible k F {k o -self-dual monic polynomials in k F rXs, and a piq P " 1 2 m piq P pm piq P`1 q. Since the degree of each such polynomial P is odd, we have Thus, if one m piq P is 1, 2 pmod 4q and the other is 0, 3 pmod 4q, then m p1q P cannot be exchanged with m p2q P independently of other changes. This is exactly reflected in the (expected) size of the L-packet as follows.
We saw in Remark 9.2 that, in this case, we have qpπq " epπq. Moreover, the formula for the reducibility points shows that P P Q 0 pπq if and only if one of s π pρ P q, s π pρ 1 P q is an odd integer; thus Q 0 pπq is empty if and only if e 0 pπq " 0. Now suppose we are given a map ε : Qpπq Ñ t1, 2u such that #tP P Q 0 pπq : εpP q " 2u is even; for irreducible k F {k o -self-dual monic P R Qpπq, we set εpP q " 1. Then we can find, for i " 1, 2, a semisimple element s In this way, we construct 2 qpπq´e0pπq inequivalent irreducible cuspidal representations of G with the same inertial reducibility set as π, which is exactly the number of cuspidal representations in the L-packet of π.
9.3. Special orthogonal and ramified unitary groups. A similar analysis can be made in the cases of special orthogonal and unitary groups G. The constraints for the maps ε are like those in the unramified case, since the anisotropic dimensions of the groups G piq Ni are determined by the group G, as is the sum of the dimensions of the spaces on which G piq Ni act. Since the details are rather similar to the cases above, we only sketch them.
We are given an irreducible cuspidal representation π of G. For P a self-dual monic polynomial in k F rXs, as in previous cases, we get integers m piq P , for i " 1, 2. We modify slightly the definition of Qpπq, replacing it with $ ' & ' %
For P a self-dual monic polynomial in k F rXs, we set f P " # 1 if degpP q " 1, 2 otherwise, and then define Q 0 pπq " ) .
Then we again constrain our map ε : Qpπq Ñ t1, 2u such that #tP P Q 0 pπq : εpP q " 2u is even. For each such ε we can construct a finite set of irreducible cuspidal representations of G. The total number of cuspidal representations obtained in this way is one, two or four times the expected number of cuspidal representations in the packet, or half this number ; the latter can occur only in the case of even orthogonal groups. We illustrate this, in particular the last case, with examples, using the same notation for the quadratic characters of GL 1 pFq as in Examples 9.3 and 9.4. Example 9.6. Let G " SOpVq be the (split) special orthogonal group of an 8-dimensional orthogonal space V with Witt index 4. Denote by J 4,0 , J 0,4 the maximal compact subgroups whose reductive quotients are SO8 pk F q. Denote by τ the unipotent irreducible cuspidal representation of SO8 pk F q, which we may inflate to either J 4,0 or J 0,4 , and thus obtain irreducible cuspidal representations π " c-Ind G J4,0 τ and π 1 " c-Ind G J0,4 τ . We have IRedpπq " IRedpπ 1 q " ttpr1s, 2q, pr1s, 2quu and the corresponding Galois parameter has the form ϕ π " ϕ π 1 " 1 b pst 3 ' st 1 q ' ω 0 b pst 3 ' st 1 q.
According to the discussion at the beginning of the section, the packet should contain four cuspidal representations, but π, π 1 are the only two cuspidal representations with this inertial reducibility set. This disparity comes from the difference between the group G and the full orthogonal group G`" OpVq. By Proposition 7.9(ii), the representation τ extends to a representation of O8 pk F q, in two ways, and inducing the inflation of these two representations from J4 ,0 and J0 ,4 to G`, we obtain four inequivalent irreducible cuspidal representations, two restricting to π and the other two to π 1 .
This example illustrates that, for even orthogonal groups, the expected number of representations in a packet should be interpreted for the full orthogonal group, rather than for the special orthogonal group.
Example 9.7. Let G " SOpVq be the special orthogonal group of a 5-dimensional orthogonal space V with Witt index 2, and denote by J 2,0 the maximal compact subgroup whose reductive quotient G 2,0 has connected component G o 2,0 » SO4 pk F qˆSO 1 pk F q. In the dual of the finite group SO4 pk F q there is an element s with characteristic polynomial pX´1q 2 pX`1q 2 such that the Lusztig series E pSO4 pk F q, sq contains a cuspidal representation τ (in fact, two such representations). The inflation of τ has two extensions to G 2,0 and we denote by λ π the inflation to J 2,0 of one such extension. Then π " c-Ind G J2,0 λ π is an irreducible cuspidal representation of G, for which IRedpπq " ttpr1s, 2q, pr1s, 1q, prω 1 s, 2q, prω 1 s, 1quu, and the corresponding Galois parameter ϕ π has the form ϕ π " ω b st 2 'ω 1 ω 1 b st 2 , for some unramified characters ω, ω 1 of order at most 2. For each choice of ω, ω 1 , the corresponding packet should contain a unique cuspidal representation, while we have constructed four such representations. Thus we have the irreducible cuspidal representations in a union of four L-packets.
Example 9.8. Let G " SOpVq be the special orthogonal group of a 20-dimensional orthogonal space V with Witt index 8 and anisotropic part of dimension 4. Denote by J 4,4 the maximal compact subgroup whose reductive quotient G 4,4 has connected component G o 4,4 » SO1 0 pk F qˆSO1 0 pk F q. In the dual of the finite group SO1 0 pk F q there are elements s 1 , s 2 with characteristic polynomials pX´1q 8 pX`1q 2 and pX´1q 2 pX`1q 8 respectively, and such that the corresponding Lusztig series E pSO1 0 pk F q, s i q contains a cuspidal representation τ i (in fact, two such representations). The representation τ 1 b τ 2 has two extensions to G 4,4 and we denote by λ π the inflation to J 4,4 of one such extension. Then π " c-Ind G J4,4 λ π is an irreducible cuspidal representation of G, for which IRedpπq " ttpr1s, 3q, pr1s, 1q, prω 1 s, 3q, prω 1 s, 1quu.
From the two choices for each of τ 1 , τ 2 above, and the two choices of extension to G 4,4 , we get 8 representations. However, we also get 8 more by exchanging the roles of τ 1 , τ 2 , and these also have the same inertial reducibility (multi)set. However, each of these irreducible cuspidal representations has two extensions to the full orthogonal group G`. Thus we in fact have the irreducible cuspidal representations in the union of four Lpackets for the full orthogonal group G`.
In this case, there are also 16 other representations of the split special orthogonal group H " SOpV 1 q, where V 1 is a 20-dimensional orthogonal space with Witt index 10, with the same inertial reducibility set, obtained as follows. We denote by J 8,2 a maximal compact subgroup of H whose reductive quotient has connected component SO1 6 pk F qŜ O4 pk F q. In the duals of the isotropic finite groups SO1 6 pk F q and SO4 pk F q there are elements s 1 , s 2 respectively, with characteristic polynomials pX´1q 8 pX`1q 8 and pX1 q 2 pX`1q 2 respectively, such that the corresponding Lusztig series contain cuspidal representations τ 1 , τ 2 respectively (two such representations in each series). The representation τ 1 b τ 2 has two extensions to the reductive quotient of J 8,2 and, inflating and then inducing to G, we obtain an irreducible cuspidal representation. Since we can also inflate to J 2,8 , we obtain 16 inequivalent representations in this way. Again, each of these representations extends in two ways to the full orthogonal group H`. Example 9.9. Let F{F o be a ramified quadratic extension and let G " UpVq be the unitary group of a 14-dimensional hermitian space V with Witt index 6 and anisotropic part of dimension 2. Denote by J 0,6 the maximal compact subgroup whose reductive quotient G 0,6 has connected component G o 0,6 » SO2 pk F qˆSp 12 pk F q. We fix an irreducible self-dual monic polynomial P P k F rXs of degree two. Then there are semisimple elements s 1 , s 2 in the dual groups of SO2 pk F q, Sp 12 pk F q respectively, with characteristic polynomials P pXq, P pXq 6 respectively, such that the corresponding Lusztig series contain unique irreducible cuspidal representations τ 1 , τ 2 respectively. Then there is a unique irreducible representation λ π of J 0,6 inflated from a representation of G 0,6 containing τ 1 b τ 2 (since τ 1 does not extend to O2 pk F q), and π " c-Ind G J0,6 λ π is irreducible and cuspidal. We have IRedpπq " ttprρ P s, 5{2q, prρ P s, 1quu, where ρ P , ρ 1 P are the self-dual irreducible cuspidal representations of GL 2 pFq corresponding to P , and the corresponding Galois parameter has the form ϕ π " ϕ P b pst 4 ' st 2 q ' ϕ 1 P , where ϕ P , ϕ 1 P are the Galois parameters corresponding to ρ P , ρ 1 P respectively. The corresponding packet should contain a unique irreducible cuspidal representation, which is π.
As in Example 9.8, we also find an irreducible cuspidal representation of the 14dimensional quasi-split ramified unitary group with the same inertial irreducibility (multi)set, by exchanging the characteristic polynomials of s 1 , s 2 .