THE LANGLANDS PARAMETER OF A SIMPLE SUPERCUSPIDAL REPRESENTATION: ODD ORTHOGONAL GROUPS

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1 THE LANGLANDS PARAMETER OF A SIMPLE SUPERCUSPIDAL REPRESENTATION: ODD ORTHOGONAL GROUPS MOSHE ADRIAN Abstract. In this work, we explicitly compute a certain family of twisted gamma factors of a simple supercuspidal representation π of a p-adic odd orthogonal group. These computations, together with analogous computations for general linear groups carried out in previous work with Liu [AL4], allow us to give a prediction for the Langlands parameter of π. We then prove that our prediction is correct if p is sufficiently large. Contents. Introduction Acknowledgements 5 2. Notation 5 3. The local functional equation for odd orthogonal groups 5 4. The simple supercuspidal representations of GL n 9 5. The simple supercuspidal representations of SO 2l The computation of Φ(W, f s ) 2 7. The computation of Φ (W, f s ) 4 8. Langlands correspondence 7 References 9. Introduction Let G be a connected reductive group defined over a p-adic field F. Recently, Gross, Reeder, and Yu [GR0, RY4] have constructed a class of supercuspidal representations of G = G(F ), called simple supercuspidal representations. These are the supercuspidal representations of G of minimal positive depth. Date: September 3, Mathematics Subject Classification. Primary S37, 22E50; Secondary F85, 22E55. Key words and phrases. Simple supercuspidal, Local Langlands Conjecture.

2 2 MOSHE ADRIAN Of interest is the Langlands correspondence for simple supercuspidal representations of G. Recall that the Langlands correspondence is a certain conjectural finite-to-one map Π(G) Φ(G) from equivalence classes of irreducible admissible representations of G to equivalence classes of Langlands parameters of G. In the past several years there has been much activity on the Langlands correspondence. DeBacker and Reeder [DR09] have constructed a correspondence for depth zero supercuspidal representations of unramified p-adic groups, and Reeder [R08] extended this construction to certain positive depth supercuspidal representations of unramified groups. Recently, under a mild assumption on the residual characteristic, Kaletha [K3] has constructed a correspondence for simple supercuspidal representations of p-adic groups, and has subsequently in [K5] extended these results to epipelagic supercuspidal representations of p-adic groups. Finally, we would like to remark that Gross, Reeder, and Yu have also studied the Langlands correspondence for epipelagic supercuspidal representations (see [GR0], [RY4]). Our present work is the first in a series of papers dedicated to explicitly determining the Langlands parameter ϕ π of a simple supercuspidal representation π of a classical group G, using the theory of gamma factors. Let π and τ be a pair of irreducible generic representations of G and GL n, where G is either SO 2l, SO 2l+ or Sp 2l. Here, we assume that SO 2l+ is split and SO 2l is quasi-split. Fix a nontrivial character ψ of F. The Rankin-Selberg integral for G GL n was constructed, in different settings, in a series of works including [GPSR87, G90, GPSR97, GRS98, Sou93]. If G is orthogonal, we denote this Rankin-Selberg integral by Φ(W, f s ), and Φ(W, φ, f s ) otherwise. Here, W is a Whittaker function for π, f s is a certain holomorphic section of a principal series (induced from τ) depending on the complex parameter s, and φ is a Schwartz function. Applying a standard normalized intertwining operator M (τ, s) to f s, one obtains a similar integral Φ (W, f s ) = Φ(W, M (τ, s)f s ) if G is orthogonal (and Φ (W, φ, f s ) = Φ(W, φ, M (τ, s)f s ) otherwise) related to Φ(W, f s ) (or Φ(W, φ, f s )) by a functional equation γ(s, π τ, ψ)φ(w, f s ) = Φ (W, f s ) if G = SO 2l+ γ(s, π τ, ψ)φ(w, f s ) = c(s, l, τ, γ)φ (W, f s ) if G = SO 2l γ(s, π τ, ψ)φ(w, φ, f s ) = Φ (W, φ, f s ) if G = Sp 2l

3 LANGLANDS PARAMETER FOR SIMPLE SUPERCUSPIDALS 3 For an accessible paper discussing all of these functional equations, we refer the reader to the recent work of Kaplan [K4ii]. The term γ(s, π τ, ψ) is known as the gamma factor of π τ, and is the key ingredient in determining ϕ π in this paper. To determine ϕ π, one must locate the poles of the gamma factors γ(s, π τ, ψ) where τ ranges over the supercuspidal representations of various general linear groups. The location of poles will determine the supercuspidal support of π (see [M98, 3, Conjecture 3.2]), hence will determine a specific functorial lift Π to a general linear group GL, the lift corresponding to the standard L-homomorphism L G L GL (see [ACS4, 2]). In the case of simple supercuspidal representations π of a classical group G, it is expected that the supercuspidal support of Π contains tamely ramified characters of GL and/or simple supercuspidal representations. We could then use the explicit local Langlands correspondence for simple supercuspidal representations of general linear groups (see [AL4, BH0, BH4]) to determine ϕ π. While describing the Langlands correspondence explicitly is in general a difficult task, in the simple supercuspidal setting it turns out to be tractable. We now give more details in the case of G = SO 2l+, which is the central concern of this paper. The simple supercuspidal representations of G are parameterized by two pieces of data: a choice of a uniformizer ϖ in F, and a sign. More explicitly (see 5), let χ be an affine generic character of the pro-unipotent radical I + of an Iwahori I. The choice of a uniformizer ϖ in F determines an element g χ in G which normalizes I and stabilizes χ. We can extend χ to K = g χ I + in two different ways, since gχ 2 =, and π = Ind G Kχ is simple supercuspidal. Let τ be a tamely ramified character of GL. Our main result (Theorem 7.2) is Theorem.. γ(s, π τ, ψ) = χ(g χ )τ(ϖ)q /2 s. In particular, since there are no poles in γ(s, π τ, ψ), it is expected that ϕ π is an irreducible representation of W F (see [M98, Conjecture 3.2]). To prove this, one could again locate the poles of a certain family of gamma factors. However, it turns out that in this special case, one can avoid this computation by using results of Kaletha [K5] as well as some standard results about functorial lifting of classical groups. Using these ideas, in Theorem 8.3 we prove that ϕ π is an irreducible representation of W F under the assumption that p is sufficiently large (see Remark 8.5).

4 4 MOSHE ADRIAN To summarize the method, we first note that the work of [K5] shows that the character of π is stable, and so by results of Moeglin [M4], π forms an L-packet by itself. Its corresponding Langlands parameter is therefore an irreducible representation of the Weil-Deligne group, and so transfers to a discrete series representation Π of GL 2l. Using some results of [LR03], together with Theorem., we may deduce that Π is in fact supercuspidal, showing that ϕ π is an irreducible representation of W F. We may moreover precisely describe ϕ π as follows. By the results of [AL4, BH4] (see in particular [AL4, Remark 3.8] or [BH4, Lemma 2.2, Proposition 2.2]), there exists a unique irreducible 2l-dimensional representation ϕ : W F GL(2l, C) with trivial determinant, whose gamma factor γ(s, ϕ τ, ψ) equals χ(g χ )τ(( ) l+ )τ(ϖ)q /2 s (the τ(( ) l+ ) factor is introduced here because of a normalization of the gamma factor that we will need later; see Theorem 3.) for every tamely ramified character τ (we give more details on this in 4). The Langlands parameter ϕ has been explicitly described in [AL4, BH0, BH4]. In the case that p 2l, let us briefly recall this description (here we follow [AL4]). Let ϖ E be a 2l th root of ( ) l+ ϖ, and set E = F (ϖ E ). Relative to the basis ϖ 2l E, ϖ 2l 2 E,, ϖ E, of E/F, we have an embedding ι : E GL 2l. Define a character ξ of E by setting ξ +pe = λ ι, where λ(a) := ψ(a 2 +A A 2l,2l + A ϖ 2l,), for A = (A) ij GL 2l. Moreover, we define ξ k (κ E/F F k ), where κ E/F = det(ind W F F W E ( E )), E denotes the trivial character of W E, and k F is the residue field of F. Finally, we set ξ(ϖ E ) = χ(g χ )λ E/F (ψ), where λ E/F (ψ) is the Langlands constant (see [BH06, 34.3]). Then ϕ = Ind W F W E ξ. The cases of SO 2l and Sp 2l are in some sense similar. But in these cases, if π is simple supercuspidal of such a group, poles are expected to arise (depending on p) in the computations of γ(s, π τ, ψ) when τ is a tamely ramified character of GL. Thus the situation has a different flavor, and is the subject of work in progress. We now summarize the contents of the sections of this paper. In 3, we recall the functional equation for odd orthogonal groups as in [K4ii]. In 4, we recall a construction of simple supercuspidal representations of GL n as well as their tamely ramified GL -twisted gamma

5 LANGLANDS PARAMETER FOR SIMPLE SUPERCUSPIDALS 5 factors. In 5, we give a construction of simple supercuspidal representations of odd orthogonal groups that may be used in computing the Rankin-Selberg integrals from 3. In 6 and 7, we compute a family of Rankin-Selberg integrals that allow us to explicitly compute the tamely ramified GL -twisted gamma factors of a simple supercuspidal representation of SO 2l+. The main result on the values of these gamma factors is Theorem 7.2, which allows us to predict the Langlands parameter of a simple supercuspidal representation of an odd orthogonal group. Finally, in 8, we show that if p is sufficiently large, then our prediction for the Langlands parameter is indeed the correct Langlands parameter. Acknowledgements. The author wishes to thank the referee for a very thorough reading of this paper and for helping to improve it in many aspects. In particular, the referee explained to the author an argument that the functorial lift of a simple supercuspidal representation of SO 2l+ to GL 2l must be supercuspidal (see Theorem 8.3). This paper has also benefited from conversations with Radhika Ganapathy, Benedict Gross, Vita Kala, Tasho Kaletha, Eyal Kaplan, and Sandeep Varma. We thank them all. 2. Notation Let F be a p-adic field, with ring of integers o, maximal ideal p, and residue field k F = o/p. Let q denote the cardinality of k F, and fix a uniformizer ϖ. Throughout, we fix a nontrivial additive character ψ : F C that is trivial on p but nontrivial on o. A character η of F is said to be tamely ramified if it is trivial on + p. Given a connected reductive group G defined over F, let G = G(F ). We also let W F denote the Weil group of F. 3. The local functional equation for odd orthogonal groups In this section we recall the functional equation for odd orthogonal groups, as in [K4ii, 3.]. Let J r denote the r r matrix.... Let J 2l+ = diag(i l, 2, I l )J 2l+. We define (3.) SO 2l+ = { g GL 2l+ : det(g) =, t gj 2l+g = J 2l+}, (3.2) SO 2n = { g GL 2n : det(g) =, t gj 2n g = J 2n }.

6 6 MOSHE ADRIAN Let T SO2n be the diagonal split maximal torus of SO 2n. Let SO2n be the standard set of simple roots of SO 2n, so that SO2n = {ɛ ɛ 2,..., ɛ n ɛ n, ɛ n +ɛ n }, where ɛ i (t) = t i is the i-th coordinate function of t T SO2n. Let Q n be the standard maximal parabolic subgroup of SO 2n corresponding to \ {ɛ n + ɛ n }. The Levi part of Q n is isomorphic to GL n, and we denote its unipotent radical by U n. Let T SO2l+ be the split maximal torus of SO 2l+. let SO2l+ be the standard set of simple roots of SO 2l+, so that SO2l+ = {ɛ ɛ 2,..., ɛ l ɛ l, ɛ l }. The highest root is ɛ + ɛ 2. Let U SO2l+ denote the subgroup of upper triangular unipotent matrices. Let ψ be a nontrivial additive character of F. We let U GLn denote the maximal unipotent subgroup of GL n consisting of upper triangular matrices. We also denote by ψ the standard nondegenerate Whittaker character of ( n ) ψ(u) = ψ u i,i+, for u = (u ij ) U GLn. Let τ be a generic representation of GL n. We denote by W(τ, ψ ) the Whittaker model of τ with respect to ψ. For s C, define i= V SO 2n Q n (W(τ, ψ ), s) = Ind SO 2n Q n (W(τ, ψ ) det s /2 ). Thus, a function f s in V SO 2n Q n (W(τ, ψ ), s) is a smooth function on SO 2n GL n, where for any g SO 2n, m in the Levi part of Q n, and u in the unipotent radical U n, we have (3.3) f s ((mug, a) = δ /2 Q n (m) det(m) s 2 fs (g, am), and the mapping a f s (g, a) belongs to W(τ, ψ ). We now let l, n N such that l n. Let U SO2l+ denote the subgroup of upper triangular unipotent matrices in SO 2l+. We use the notation ψ again to define a non-degenerate character on U SO2l+ by ( l ) ψ(u) = ψ u i,i+ + 2 u l,l+. i= We will also need the standard non-degenerate character on U SO2l+ : ( l ) ψ std (u) = ψ u i,i+. i=

7 LANGLANDS PARAMETER FOR SIMPLE SUPERCUSPIDALS 7 Let π be a generic representation of SO 2l+ with respect to ψ. We shall denote by W(π, ψ) the Whittaker model of π with respect to ψ. Let τ be a generic representation of GL n with respect to ψ. In computing γ-factors, we will be interested in the following local integrals (see [K4ii]): Φ(W, f s ) := U SO2n \SO 2n X (n,l) W (xj n,l (h))f s (h, I n )dxdh, where W W(π, ψ), f s V SO 2n Q n (W(τ, ψ ), s), and where dh is a fixed right-invariant Haar measure on the quotient space U SO2n \ SO 2n, and dx is a fixed Haar measure. Here, j n,l : SO 2n SO 2l+ is the map ( ) A B A B I C D 2(l n)+ SO 2l+, C D I n y I l n X (n,l) = I l n SO 2l+. y Note that y F l is a column vector, and y F l is a row vector. Moreover, the notation y means that these coordinates are determined by the coordinates of y, according to the matrix defining SO 2l+. Explicitly, if y is the transpose of the row vector (y, y 2,...y l n, y l n ), then y is the row vector ( y l n, y l n,..., y 2, y ). Now let n be odd. Define a matrix ω = n 0 0 We define an intertwining operator I n I n. by M(τ, s) : V SO 2n Q n (W(τ, ψ ), s) V SO ω 2n Q n (W(τ, ψ), s) M(τ, s)f s (h, a) = f s (uwn h, a )du, U n

8 8 MOSHE ADRIAN where ( ) I w n = n I n I 2(n ) SO 2n, and where U n is the opposite unipotent radical to U n. Here, τ (g) := τ(g ), where g := J n t g J n. We note that in the case that n = (the case that we will be interested in throughout the paper), we have that τ (g) = τ(g ), and also that M(τ, s)f s (h, a) = f s (h, a). Then we will also be interested in the integrals (see [K4ii]) Φ (W, f s ), where Φ (W, f s ) := γ(2s, τ, 2, ψ) Moreover, U SO2n \SO 2n X (n,l) W (ĉ n,l xj n,l (h)δ o ω )M(τ, s)f s ( ω h, b n )dxdh. ĉ n,l = diag(i n, I l n,, I l n, I n ) SO 2l+, δ o = diag(i l,, I l ), b n = diag(,,...,, ) GL n, I n ω = I 2(l n)+ I n. Finally, γ(2s, τ, 2, ψ) is defined using the Langlands-Shahidi method (see [K4ii, 3.]. We then have a functional equation for odd orthogonal groups (see [K4ii, 4]. (3.4) γ(s, π τ, ψ)φ(w, f s ) = Φ (W, f s ). It was then shown that in [K4ii] that if one suitably normalizes γ(s, π τ, ψ), then we can get a gamma factor that is equal to that of Shahidi. Theorem 3.. [K4ii, 4, Corollary ] If we define Γ(s, π τ, ψ) = ω π ( ) n ω τ ( ) l γ(s, π τ, ψ), then Γ(s, π τ, ψ) coincides with Shahidi s gamma factor for π τ defined in [Sha90]. Remark 3.2. We will be interested in the case that n =. This is permissible (see [K4ii, 3.]).

9 LANGLANDS PARAMETER FOR SIMPLE SUPERCUSPIDALS 9 4. The simple supercuspidal representations of GL n In this section we review the definition of a simple supercuspidal representation of GL n, as in [AL4] (see also [KL5]). We then recall the values of a family of its twisted gamma factors, in order to predict the Langlands parameter of a simple supercuspidal of SO 2l+. Let GL n = GL n (F ), Z the center of G, K = GL n (o) the standard maximal compact subgroup, I the standard Iwahori subgroup, and I + its pro-unipotent radical. Fix a nontrivial additive character ψ of F that is trivial on p and nontrivial on o. Let U denote the unipotent radical of the standard upper triangular Borel subgroup of GL n. For any u U, we recall from 3 the standard nondegenerate Whittaker character ψ of U: ( n ) ψ(u) = ψ u i,i+. Set H = ZI +, and fix a character ω of Z = F, trivial on + p. We define a character χ : H C by χ(zk) = ω(z)ψ(r + r r n + r n ) for x r x 2 r 2 k = r n ϖr n x n The compactly induced representation π χ := cind G Hχ is then a direct sum of n distinct irreducible supercuspidal representations of GL n. They are parameterized by ζ, where ζ is a complex n th root of ω(ϖ), as follows. Set g χ = 0 i=... ϖ 0 and set H = g χ H. Then the summands of π χ are σ ζ χ := cind G H χ ζ where χ ζ (g j χh) = ζ j χ(h), as ζ runs over the complex n th roots of ω(ϖ). σ ζ χ is called a simple supercuspidal representation of GL n. As we vary ϖ, ζ, and ω, we obtain all simple supercuspidal representations of GL n.

10 0 MOSHE ADRIAN Since we will need to consider the choice of uniformizer later, we denote by σ(ϖ, ζ, ω) the simple supercuspidal representation of GL n determined by the triple (ϖ, ζ, ω). Finally, for the simple supercuspidal representation σ ζ χ, we may define a Whittaker function on GL n by setting { ψ(u)χζ (h W (g) = ) if g = uh UH 0 else This function is well-defined, by definition of ψ and χ ζ. Moreover, W Ind G U ψ cindg H χ ζ. Using this Whittaker function, it turns out to be not so difficult to compute the Rankin-Selberg integrals that arise in the definition of the twisted gamma factors γ(s, σχ ζ τ, ψ), where τ is a tamely ramified character of F. In 5, we will define an analogous Whittaker function for a simple supercuspidal representation π of SO 2l+, which will allow us to compute the relevant Rankin-Selberg integrals for π. In [AL4] (see also [BH4, Lemma 2.2]), we explicitly computed γ(s, σχ ζ τ, ψ): Lemma 4.. [AL4, Lemma 3.4] Let τ be a tamely ramified character of GL. Then γ(s, σ ζ χ τ, ψ) = τ( ) n τ(ϖ)χ(g χ )q /2 s 5. The simple supercuspidal representations of SO 2l+ In this section we explicitly construct the simple supercuspidal representations of SO 2l+, following [RY4, 2]. We then define an explicit Whittaker function for each such simple supercuspidal. Let T = T SO2l+ denote the diagonal split maximal torus of SO 2l+. Associated to T we have the set of affine roots Ψ. Let X (T ) denote the character lattice of T, let T 0 be the maximal compact subgroup of T, and set T = t T 0 : λ(t) + p λ X (T ). To each α Ψ we have an associated affine root group U α in G. Associated to the standard set of simple roots SO2l+ we have a set of simple affine Π and positive affine roots Ψ +. Set I = T 0, U α : α Ψ +, I + = T, U α : α Ψ +, I ++ = T, U α : α Ψ + \ Π.

11 LANGLANDS PARAMETER FOR SIMPLE SUPERCUSPIDALS In terms of the Moy-Prasad filtration, if x denotes the barycenter of the fundamental alcove, then I = G x = G x,0, I + = G x+ = G x,, and 2l I ++ = G x, +. 2l For (t, t 2,..., t l+ ) o /( + p) o /( + p) o /( + p), we define a character χ : I + C h 2l, h ψ(t h 2 + t 2 h t l h l,l + t l x l,l+ + t l+ ϖ ) for h = (h) ij I +. These χ s are called the affine generic characters of I + (see [RY4, 2.6] and [GR0, Lemma 9.2] for more details on this). The orbits of affine generic characters are parameterized by the set of elements in o /( + p), as follows. T SO 2l+ (o) normalizes I +, so acts on the set of affine generic characters by conjugation. Every orbit of affine generic characters contains one of the form (,,...,, t), for t o /( + p). Specifically, after conjugating by diag((t t 2 t 3 t l ), (t 2 t 3 t l )..., t l,, t l,..., t 2 t 3 t l, t t 2 t 3 t l ), the orbit of (t, t 2,..., t l+ ) contains (,,...,, t), where t = t l+ t. t 2 2 t2 3 t2 l Instead of viewing the affine generic characters as parameterized by t o /( + p), we will set t = and let the affine generic characters be parameterized by the various choices of uniformizer in F. Since we have already fixed an (arbitrary) uniformizer ahead of time in 2, we have therefore fixed an affine generic character χ. The compactly induced representation π χ := ind G Hχ is a direct sum of 2 distinct irreducible supercuspidal representations of SO 2l+. They are parameterized ζ {, }, as follows. Set g χ = ϖ I 2l ϖ One can compute that g χ normalizes I +, and that χ(g χ hgχ ) = χ(h) h I +, as follows. We first note that χ(g χ hg χ ) = ψ( ϖ h 2l+,2 + h 23 + h h l,l+ h 2l,2l+ ),. χ(h) = ψ(h 2 + h 23 + h h l,l+ + ϖ h 2l, ). By explicit computation inside the group SO 2l+, one then shows that h 2l+,2 = h 2l, and h 2l,2l+ = h 2. Set H = g χ I +. Since g 2 χ =, we may extend χ to a character of H in two different ways by mapping g χ to ζ {, }. Then the

12 2 MOSHE ADRIAN summands of π χ are the compactly induced representations π ζ χ := ind G H χ ζ where χ ζ (g χ h) = ζ χ(h), where ζ {, } and h I +. πχ ζ is called a simple supercuspidal representation of SO 2l+. As we vary ϖ and ζ, we obtain all simple supercuspidal representations of GL n. Moreover, if one fixes pairs (ϖ, ζ ) and (ϖ 2, ζ 2 ), where ϖ i are uniformizers and ζ i {, }, then the associated simple supercuspidals π, π 2 of SO 2l+ are isomorphic if and only if ϖ ϖ 2 mod ( + p) and ζ = ζ 2. Since we will need to consider the choice of uniformizer later, we denote by π(ϖ, ζ) the simple supercuspidal representation of SO 2l+ determined by the pair (ϖ, ζ). To compute the Rankin-Selberg integrals, we need a Whittaker ( function for πχ ζ with respect to ψ. We note that χ ζ (u) = l ) ψ i= u i,i+ for u U SO2l+ I +, so the function (5.) { ψstd (u)χ W std (g) := ζ (gχ)χ i ζ (k) if g = ugχk i U g SO2l+ χ I + 0 else is well-defined and lives in Ind G U ψ std ind G H χ ζ, so is therefore a Whittaker function for π ζ χ with respect to ψ std. In order to compute the Rankin-Selberg integrals Φ, Φ, we need a Whittaker function for π ζ χ with respect to ψ. For this, note that there is a canonical isomorphism Ind G Uψ std Ω Ind G Uψ f γ f, where γ f(g) := f(γgγ ), and γ = diag(i l, 2,, 2, I l ). The function W := γ W std is therefore a Whittaker function for the representation Ω(π ζ χ) with respect to ψ. Since π ζ χ and Ω(π ζ χ) are isomorphic as representations, we will also say that W is a Whittaker function for π ζ χ with respect to ψ. 6. The computation of Φ(W, f s ) In this section, we compute one side of the functional equation (3.4) that defines γ(s, π τ, ψ), where π is a simple supercuspidal representation of SO 2l+, and τ is a tamely ramified character of GL. We first note that since τ is a character of GL, Φ(W, f s ) simplifies to W (xj,l (h))f s (h, )dxdh, X (,l) U SO2 \SO 2

13 LANGLANDS PARAMETER FOR SIMPLE SUPERCUSPIDALS 3 Moreover, note that by (3.2), {( ) } z 0 SO 2 = 0 z : z F = GL. We will pass between SO 2 and GL when convenient. We now define f s V SO 2 Q (W(τ, ψ ), s) (see (3.3)) by (6.) f s (h, a) = det(h) s /2 τ(ah), where τ(h) and det(h), mean τ and det, applied to the upper left entry of h. We remind the reader that Q = SO2, so in particular δ Q. Recall the Whittaker function W that we defined earlier (see 5) for πχ ζ = Ind SO 2l+ g χ I χ + ζ. We will need the following computation more than once: if u = (u) ij U SO2l+ and k = (k) ij I +, we get that (6.2) ug χ k = ϖk ϖk 2 ϖk n Now let y I l x = I l X,l, y with y the transpose of the row vector (y, y 2,..., y l 2, y l ) (so that y = ( y l, y l 2,..., y 2, y )). Lemma 6.. If W (xj n,l (h)) 0, then y i p for i =, 2,..., l 2, y l 2p, and a + p, where h = diag(a, a ). In particular, x I ++. Proof. Let h = diag(a, a ) SO 2. Note that W (xj n,l (h)) = W std (γxj n,l (h)γ ). Then γxj n,l (h)γ embeds in SO 2l+ as a w I l I l. w a Here, w is the transpose of the row vector (ay, ay 2,..., ay l 2, 2 ay l ), and w = ( 2 y l, y l 2, y l 3,..., y 2, y ). By comparing lower left matrix entries, one can see that γxj n,l (h)γ / U SO2l+ g χ I + (see (6.2)).

14 4 MOSHE ADRIAN Moreover, by comparing bottom right matrix entries, one can see that if γxj n,l (h)γ U SO2l+ I +, we must have that a + p. Finally, by comparing bottom rows, that y i p for i =, 2,..., l 2, and that y l 2p. We may now compute Φ(W, f s ). Corollary 6.2. U SO2 \SO 2 X (,l) W (xj,l (h))f s (h, )dxdh = vol(p) l 2 vol(2p)vol( + p). Proof. By Lemma 6., since U SO2 =, and since τ is tamely ramified, we obtain W (xj,l (h))f s (h, )dxdh U SO2 \SO 2 X (,l) = W (xj,l (h))dxdh. +p X (,l) Moreover, because of the constraints placed on the y i in the statement of Lemma 6., and by definition of W, this last double integral equals vol(p) l 2 vol(2p)vol( + p), as claimed. 7. The computation of Φ (W, f s ) In this section, we compute Φ (W, f s ), for the same choice of W and f s as in 6. Since τ is a character of GL, Φ (W, f s ) simplifies to Φ (W, f s ) = γ(2s, τ, 2, ψ) U SO2 \SO 2 We note in particular that Now let ω = y I l x = X (,l) W (ĉ,l xj,l (h)δ o ω )M(τ, s)f s (h, )dxdh. I 2l I l. X,l, y with y the transpose of the row vector (y, y 2,..., y l 2, y l ) (so that y = ( y l, y l 2,..., y 2, y ).

15 LANGLANDS PARAMETER FOR SIMPLE SUPERCUSPIDALS 5 Lemma 7.. In order that W (ĉ,l xj,l (h)δ o ω ) 0, it must be that y i p for i =, 2,..., l 2, that y l 2p, and that a ϖ ( + p), where h = diag(a, a ). In particular, x I +. Proof. Note first that W (ĉ,l xj,l (h)δ o ω ) = W std (γĉ,l xj,l (h)δ o ω γ ). If h = diag(a, a ), then a 0 I l 0 0 w γĉ,l xj,l (h)δ o ω γ = I l 0, a 0 0 w 0 where w is the transpose of the row vector ( ay, ay 2,..., ay l 2, ay 2 l ) and w = ( y 2 l, y l 2, y l 3,..., y 2, y ). We now perform some matrix computations. In order that W std (γĉ,l xj,l (h)δ o ω γ ) 0, we must have that γĉ,l xj,l (h)δ o ω γ U SO2l+ g χ I +, by definition of W std. We first consider the double coset U SO2l+ g χ I +. Namely, suppose that γĉ,l xj,l (h)δ o ω γ = ug χ k, for some u U SO2l+, k I +. Let us compare the last row of γĉ,l xj,l (h)δ o ω γ to the last row of ug χ k. In particular, since k + p, we must have that a ϖ ( + p) (see (6.2)). Since k j o for j 2, we moreover require that y i p for i =, 2,..., l 2, and y l 2p. A similar type of analysis shows that γĉ,l xj,l (h)δ o ω γ / U SO2l+ I +. We now have our main result. Theorem 7.2. γ(s, π ζ χ τ, ψ) = χ ζ (g χ )τ(ϖ)q /2 s. Proof. We first note that since τ is a character of GL, we have that γ(2s, τ, 2, ψ) =. This can be explicitly calculated from its definition, which can be found in [K4ii, 3.]. Now let h = diag(a, a ) SO 2, and suppose that W std (γĉ,l xj,l (h)δ o ω γ ) 0. By Lemma 7., we have that a ϖ ( + p). In particular, we can write a = ϖ v, for some v + p. We now decompose γĉ,l xj,l (h)δ o ω γ in the double coset U SO2l+ g χ I + as

16 6 MOSHE ADRIAN a 0 I l 0 0 w γĉ,l xj,l (h)δ o ω γ = I l 0 a 0 0 w 0 ϖ v ϖ w I l w = I 2l I l, ϖ v where w, w are as in Lemma 7.. This decomposition implies by Lemma 7. that if W std (γĉ,l xj,l (h)δ o ω γ ) 0, then in fact W std (γĉ,l xj,l (h)δ o ω γ ) = χ ζ (g χ ). Finally, one can see that M(τ, s)f s (h, ) = f s (h, ). Since a = ϖv, we have f s (h, ) = det(ϖv ) s /2 τ(ϖv ). Therefore, because of the constraints placed on the y i in the statement of Lemma 7., we have Φ (W, f s ) = W (ĉ,l xj,l (h)δ o ω )M(τ, s)f s (h, )dxdh SO 2 X (,l) = vol(p) l 2 vol(2p) χ ζ (g χ ) det(ϖv ) s /2 τ(ϖv )dh +p = vol(p) l 2 vol(2p)χ ζ (g χ ) det(ϖ) s /2 τ(ϖ)vol( + p) = vol(p) l 2 vol(2p)χ ζ (g χ )q /2 s τ(ϖ)vol( + p), since τ is tamely ramified. Combining this computation with Corollary 6.2 and (3.4), we have our result. Corollary 7.3. Γ(s, π ζ χ τ, ψ) = χ ζ (g χ )τ( ) l τ(ϖ)q /2 s. We now compare Γ(s, πχ ζ τ, ψ) with the gamma factors of simple supercuspidal representations of GL 2l. First, we note that we may rewrite πχ ζ as π(ϖ, ζ) (see 5). By Corollary 7.3 and Lemma 4., we obtain the following matching of gamma factors. Corollary 7.4. The representation σ(( ) l+ ϖ, ζ, ) of GL 2l and the representation π(ϖ, ζ) of SO 2l+ share the same gamma factors twisted

17 LANGLANDS PARAMETER FOR SIMPLE SUPERCUSPIDALS 7 by tamely ramified characters of GL. Namely, Γ(s, π(ϖ, ζ) τ, ψ) = γ(s, σ(( ) l+ ϖ, ζ, ) τ, ψ), for all tamely ramified characters τ of GL. 8. Langlands correspondence In this section we determine the Langlands parameter of a simple supercuspidal representation of SO 2l+, under the assumption that p is sufficiently large (see Remark 8.5). We first need to recall the definition of local transfer. Definition 8.. [ACS4, Definition 3.] Let π be an irreducible generic representation of SO 2l+. Let Π be an irreducible representation of GL 2l. We say that Π is a local transfer of π if L(s, π ρ) = L(s, Π ρ) and γ(s, π ρ, ψ) = γ(s, Π ρ, ψ) for all irreducible, unitary, supercuspidal representations ρ of GL m, m 2l. The L, γ-factors on the left hand side are those of the Langlands-Shahidi method while those on the right hand side are defined via either the Rankin-Selberg method [JPSS83] or the Langlands-Shahidi methods [Sha90]. Remark 8.2. It is not true in general that Rankin-Selberg factors for GL GL equal the corresponding Langlands-Shahidi gamma factors. Indeed, this is the crux of Shahidi s paper [Sha84]. However, one can show that if a generic representation of a GL arises as a local lift from a classical group, then its twisted Rankin-Selberg gamma factors equal its twisted Langlands-Shahidi gamma factors. Recall the Langlands parameter Ind W F W E ξ constructed in. We may now prove: Theorem 8.3. For p sufficiently large, the functorial lift Π of π = π ζ χ (under the embedding Sp(2l, C) GL(2l, C)) to GL 2l is supercuspidal. Proof. We need to recall some results of [K5]. Specialized to the group G = SO 2l+, the work of [K5] implies that the character of π is stable. To see this, let φ be the Langlands parameter constructed in [K5] associated to π. The part of the L-packet that Kaletha constructs, on the group SO 2l+, is in bijection with the characters of the centralizer S φ of φ that are trivial on the center of Ĝ = Sp 2l(C) (see [K5, (0.0.)]). In this case the centralizer of φ is the subgroup of the maximal torus ˆT of Ĝ consisting of those elements that are fixed by a Coxeter element w ([K5, p. 29, 34]). A computation shows that ( T ) w = Z(Ĝ).

18 8 MOSHE ADRIAN Therefore, the quotient S φ /Z(Ĝ) is trivial. A special case of [K5, Theorem 5.2.] now shows that the character of π is stable. Therefore, by [M4, Corollary 4.5, Theorem 4.7], we have that π constitutes an entire L-packet by itself. Since π constitutes an L-packet, the component group of its associated Langlands parameter (the parameter of Arthur [A3]) ϕ π is trivial, so that ϕ π is an irreducible 2l dimensional representation of the Weil- Deligne group. The associated representation Π of GL 2l via the local Langlands correspondence is therefore a discrete series representation. We now note that the functorial lift Π (which comes from Arthur s work) agrees with the lifting constructed by [CKPSS04]. Therefore, by Definition 8., we in particular have that γ(s, Π, ψ) = Γ(s, π, ψ). Since the conductor of Π can be read off γ(s, Π, ψ) (see, in particular, [BH4,.3]), we conclude that the conductor of Π is 2l+ by Corollary 7.3. We now turn to some results of [LR03]. In particular, we follow some arguments in the proof of [LR03, Theorem 3.]. According to the classification of Bernstein and Zelevinskii, Π is equivalent to the Langlands Quotient Q(λ) of the parabolic induction Ind H P (σ). Here, P is a standard parabolic subgroup of H = GL 2l whose Levi subgroup is of the form GL a... GL a, and σ is a representation of P of the form λ λ det... λ det b, where λ is a supercuspidal representation of GL a. Note that ab = 2l. There are two cases to consider. If a = and λ is unramified, then Π is an unramified twist of the Steinberg representation. But an unramified twist of the Steinberg representation has conductor 2l, a contradiction. Otherwise, denoting c by conductor, the proof of [LR03, Theorem 3.] further shows that c(π) = c(λ)b. Together with ab = 2l and c(π) = 2l + we obtain immediately that b =, so that a = 2l, so that Π = λ is supercuspidal. Corollary 8.4. For p sufficiently large, the Langlands parameter of π = π ζ χ is Ind W F W E ξ. Proof. We have concluded in Theorem 8.3 that the functorial lift Π of π to GL(2l, F ) is supercuspidal. Π has conductor 2l +, so in fact Π is simple supercuspidal. We also note that this lift must have trivial central character since it is a lift from SO 2l+. By Corollary 7.4 and by [AL4, Remark 3.8] (see also [BH4, Lemma 2.2, Proposition 2.2]), this lift must be σ(( ) l+ ϖ, ζ, ). The main result of [AL4, 3] says that the Langlands parameter of σχ ζ is Ind W F W E ξ.

19 LANGLANDS PARAMETER FOR SIMPLE SUPERCUSPIDALS 9 Remark 8.5. The meaning of p sufficiently large is as follows. Let e be the ramification index of F/Q p. Kaletha proves the stability results needed in Corollary 8.3 under the assumption that p (2 + e)(2l + ) [K5, 5.]. References [AL4] M. Adrian and B. Liu, Some results on simple supercuspidal representations of GL n (F ). Submitted [A3] J. Arthur, The endoscopic classification of representations: orthogonal and symplectic groups. Colloquium Publications, vol. 6, American Mathematical Society, 203. [ACS4] M. Asgari, J. Cogdell, and F. Shahidi Local Transfer and Reducibility of Induced Representations of p-adic Groups of Classical Type Cotemp. Math., AMS. To appear., 204. [BH06] C. Bushnell and G. Henniart, The Local Langlands Conjecture for GL(2). A Series of Comprehensive Studies in Mathematics, Volume 335, Springer Berlin Heidelberg, [BH0] C. Bushnell and G. Henniart, The essentially tame local Langlands correspondence, III: the general case. Proc. Lond. Math. Soc. (3) 0 (200), no. 2, [BH4] C. Bushnell and G. Henniart, Langlands parameters for epipelagic representations of GL n. Math. Ann. 358 (204), no. -2, [CKPSS04] J. Cogdell, H.H. Kim, I.I. Piatetski-Shapiro, and F. Shahidi. Functoriality for the classical groups. Publ. Math. IHES. 99 (2004), [DR09] S. DeBacker and M. Reeder, Depth-zero supercuspidal L-packets and their stability. Ann. of Math. (2) 69 (2009), no. 3, [GPSR87] S. Gelbart, I.I. Piatetski-Shapiro, and S. Rallis, L-functions for G GL(n). Lecture Notes in Math., vol. 254, Springer-Verlag, New York, 987. [G90] D. Ginzburg, L-functions for SO n GL k. J. Reine Angew. Math. 405 (990) [GPSR97] D. Ginzburg, I.I. Piatetski-Shapiro, and S. Rallis, L-functions for the Orthogonal Group. Mem. Amer. Math. Soc., vol. 6, Amer. Math. Soc., Providence, RI, 997. [GRS98] D. Ginzburg, S. Rallis, and D. Soudry, L-functions for symplectic groups. Bull. Soc. Math. France 26 (998) [GR0] B. Gross and M. Reeder, Arithmetic invariants of discrete Langlands parameters. Duke Math. J. 54 (200), no. 3, [JPSS83] H. Jacquet, I. Piatetskii-Shapiro and J. Shalika, Rankin Selberg convolutions. Amer. J. Math. 05 (983), [K4i] V. Kala Density of Self-Dual Automorphic Representations of GL n (A Q ). Ph.D. Thesis, Purdue University, 204. [K3] T. Kaletha, Simple wild L-packets. J. Inst. Math. Jussieu 2 (203), no., [K5] T. Kaletha, Epipelagic L-packets and rectifying characters. Invent. Math., to appear.

20 20 MOSHE ADRIAN [K4ii] E. Kaplan Complementary results on the Rankin-Selberg gamma factors of classical groups. J. Number Theory (204), [KL5] A. Knightly and C. Li, Simple supercuspidal representations of GL(n). Taiwanese J. Math., to appear. [LR03] J. Lansky and A. Raghuram, On the correspondence of representations between GL(n) and division algebras. Proc. Amer. Math. Soc. 3 (2003), no. 5, [M4] C. Moeglin, Paquets stables des sries discrtes accessibles par endoscopie tordue; leur paramtre de Langlands. Automorphic forms and related geometry: assessing the legacy of I. I. Piatetski-Shapiro, , Contemp. Math., 64, Amer. Math. Soc., Providence, RI, 204. [M98] G. Muic Some Results on Square Integrable Representations; Irreducibility of Standard Representations. IMRN, No. 4 (998), pages [R08] M. Reeder, Supercuspidal L-packets of positive depth and twisted Coxeter elements. Crelle s Journal, 620, (2008), pages 33. [RY4] M. Reeder and J.-K Yu, Epipelagic representations and invariant theory. Journal of the AMS, 27 (204), pages [Sha84] F. Shahidi, Fourier Transforms of Intertwining Operators and Plancherel Measures for GL(n). American Journal of Mathematics. 06 (984), No., p. 67. [Sha90] F. Shahidi, A proof of Langlands conjecture on plancherel measures; complementary series for p-adic groups. Ann. of Math. 32 (990), [Sou93] D. Soudry, Rankin-Selberg Convolutions for SO 2l+ GL n : Local Theory. Mem. Amer. Math. Soc., vol. 500, Amer. Math. Soc., Providence, RI, 993. Department of Mathematics, Queens College, CUNY, Queens, NY address: moshe.adrian@qc.cuny.edu