On the metaplectic analog of Kazhdan’s “endoscopic” lifting (original) (raw)
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Lifting of characters on orthogonal and metaplectic groups
Duke Mathematical Journal, 1998
An important principle in representation theory and automorphic forms is that of lifting or transfer of representations between reductive algebraic groups. Endoscopic transfer and base change are primary examples. Another type of example is provided by theta-lifting between members of a reductive dual pair. In this paper we study lifting, defined directly on characters, between special orthogonal groups SO(2n + 1) over R and the non-linear metaplectic group Sp(2n, R). This is closely related both to endoscopy and theta-lifting, and is an aspect of the duality between root systems of type B n and C n. Let π be an irreducible representation of SO(p, q), the special orthogonal group of a symmetric bilinear form in p + q = 2n + 1 real variables; π has a non-zero theta-lift to a representation π ′ of Sp(2n, R). A natural question is: what is the relationship, if any, between the global characters of π and π ′ ? When n = 1 this is closely related to the Shimura correspondence, which has been the subject of extensive study. Evidence for such a relation is provided by the orbit correspondence, which provides a matching of semisimple conjugacy classes of SO(p, q) and Sp(2n, R). This is analogous to the matching of stable conjugacy classes in the theory of endoscopy. In fact there is a natural bijection between (strongly) regular, semisimple, stable conjugacy classes in the split groups SO(n + 1, n) and Sp(2n, R). In elementary terms two such conjugacy classes correspond if they have the same non-trivial eigenvalues. Alternatively there is a bijection between conjugacy classes of Cartan subgroups in these two groups. The main ideas are best illustrated by the example of the discrete series. Let π SO (λ) be a discrete series representation of SO(n+1, n). We have fixed a compact Cartan subgroup T , and λ ∈ t * is a Harish-Chandra parameter. In the usual coordinates λ = (a 1 ,. .. , a k ; b 1 ,. .. , b ℓ) (1.1)(a) with a i , b j ∈ Z + 1 2 , a 1 > • • • > a k > 0, b 1 > • • • > b ℓ > 0. Fix a compact Cartan subgroup T ′ of Sp(2n, R), with inverse imageT ′ in Sp(2n, R). The theta-lift of π SO (λ) to Sp(2n, R) is the discrete series representation π Sp (λ ′) with Harish-Chandra parameter
Journal d'Analyse Mathématique, 1988
The Selberg trace formula is of unquestionable value for the study of automorphic forms and related objects. In principal it is a simple and natural formula, generalizing the Poisson summation formula, relating traces of convolution operators with orbital integrals. This paper is motivated by the belief that such a fundamental and natural relation should admit a simple and short proof. This is accomplished here for test functions with a single supercusp-component, and another component which is spherical and "sufficiently-admissible" with respect to the other components. The resulting trace formula is then used to sharpen and extend the metaplectic correspondence, and the simple algebras correspondence, of automorphic representations, to the context of automorphic forms with a single supercuspidal component, over any global field. It will be interesting to extend these theorems to the context of all automorphic forms by means of a simple proof. Previously a simple form of the trace formula was known for test functions with two supercusp components; this was used to establish these correspondences for automorphic forms with two supercuspidal components. The notion of ~sufficienfly-admissible" spherical functions has its origins in Drinfeld's study of*he reciprocity law for GL(2) over a function field, and our form of the trace formula is analogous to Deligne's conjecture on the fixed point formula in 6tale cohomology, for a correspondence which is multiplied by a sufficiently high power of the Frobenius, on a separated scheme of finite type over a finite field. Our trace formula can be used (see [FK']) to prove the Ramanujan conjecture for automorphic forms with a supercuspidal component on GL(n) over a function field, and to reduce the reciprocity law for such forms to Deligne's conjecture. Similar techniques are used in ['F] to establish base change for GL(n) in the context of automorphic forms with a single supercuspidal component. They can be used to give short and simple proofs of rank one lifting theorems for arbitrary automorphic forms; see ["F] for base change for GL(2), [F'] for base change for U(3), and ['F'] for the symmetric square lifting from SL(2) to PGL(3). Let F be a global field, A its ring of adeles and A I the ring of finite adeles, G a connected reductive algebraic group over F with center Z. The group G of Frational points on G is discrete in the adele group G(A) of G. Put G' = G/Z and G'(A) = G(A)/Z(A). The quotient G' \ G'(A) has finite volume with respect to the unique (up to a scalar multiple) Haar measure dg on G'(A). Fix a unitary complex-valued character to of Z \ Z(A). For any place v of F let Fv be the completion of F at v, and Gv = G(F~) the group of F~-points on G. If F, is non-archimedean, let Rv denote its ring of integers. For almost all v the group G, is defined over R,, quasi-split over F~, split over an unramified extension ofF,, and t Partially supported by NSF grants.
Some Eichler-Selberg Trace Formulas
Hardy-Ramanujan Journal
The Eichler-Selberg trace formulas express the traces of Hecke operators on a spaces of cusp forms in terms of weighted sums of Hurwitz-Kronecker class numbers. For cusp forms on textrmSL_2(mathbbZ),\text {\rm SL}_2(\mathbb{Z}),textrmSL2(mathbbZ), Zagier proved these formulas by cleverly making use of the weight 3/2 nonholomorphic Eisenstein series he discovered in the 1970s. The holomorphic part of this form, its so-called {\it mock modular form}, is the generating function for these class numbers. In this expository note we revisit Zagier's method, and we show how to obtain such formulas for congruence subgroups, working out the details for Gamma0(2)\Gamma_0(2)Gamma0(2) and Gamma0(4).\Gamma_0(4).Gamma_0(4). The trace formulas fall out naturally from the computation of the Rankin-Cohen brackets of Zagier's mock modular form with specific theta functions.
Modular representations arising from self-dual $ ell$-adic representations of finite groups
Eprint Arxiv Math 9809107, 1998
Let G be a finite subgroup of a symplectic group Sp 2d (Q ℓ). Despite the fact ([3]) that G can fail to be conjugate in GL 2d (Q ℓ) to a subgroup of Sp 2d (Z ℓ), we prove that it can nevertheless be embedded in Sp 2d (F ℓ) in such a way that the characteristic polynomials are preserved (mod ℓ), as long as ℓ > 3. We start with the following "rigidity" result, which is in the spirit of similar results by Minkowski and Serre. Proposition 1. Suppose ℓ is a prime number, and K is a discrete valuation field of characteristic zero and residue characteristic ℓ. Let O denote the valuation ring and λ the maximal ideal. Let e denote the ramification index of K (i.e., ℓO = λ e), and suppose 2e < ℓ − 1. Suppose S is a free O-module of finite rank, A is an automorphism of S of finite order, and (A − 1) 2 ∈ λEnd(S). Then A = 1. Proof. This follows directly from Theorem 6.2 of [2] with n = ℓ and k = 2e. Note that the hypothesis 2e < ℓ − 1 is satisfied if e = 1 and ℓ ≥ 5. Next we state our main theorem.
On the cohomology of GL(N) and adjoint Selmer groups
arXiv (Cornell University), 2021
We prove-under certain conditions (local-global compatibility and vanishing of modulo p cohomology), a generalization of a theorem of Galatius and Venkatesh. We consider the case of GL(N) over a CM field; we construct a Hecke-equivariant injection from the divisible group associated to the first fundamental group of a derived deformation ring to the Selmer group of the twisted dual adjoint motive with divisible coefficients and we identify its cokernel as the first Tate-Shafarevich group of this motive. Actually, we also construct similar maps for higher homotopy groups with values in exterior powers of Selmer groups, although with less precise control on their kernel and cokernel. By a result of Y. Cai generalizing previous results by Galatius-Venkatesh on the graded cohomology group of a locally symmetric space, our maps relate the (non-Eisenstein) localization of the graded cohomology group for a locally symmetric space to the exterior algebra of the Selmer group of the Tate dual of the adjoint representation. We generalize this to Hida families as well. 1 2 J. TILOUINE AND E. URBAN (Gal A) There exists a continuous Galois representation ρ A : G F) → GL N (A) unramified outside np and such that for any v prime to np, char(ρ m (Frob v)) = Hecke v (X) where the Hecke polynomial at v is given by Hecke v (X) = X N − T v,1 X N −1 +. .. + (−1) i T v,i Nv i(i−1) 2 X N −i +. .. + (−1) N T v,N Nv N(N−1) 2. This conjecture is proven in [Sch15] for A = T/I where I is a nilpotent ideal of exponent bounded in terms of N and [F : Q]. The exponent of I is bounded by 4 in [NT16]. By [CGH+19], the ideal can be taken to be 0 if p splits totally in F. We assume (M IN) The level n is squarefree and ρ π is n-minimal. Write the weight λ as λ = (λ τ,i) where τ : F → Q and i = 1,. .. , N. Let S p = S F,p be the set of places of F above p. By [Ca14], it is known that the local Galois representations ρ π | G Fw is crystalline at all places w ∈ S p (see Section 2). We will assume either of the two following local conditions at p : (F L) p is unramified in F , p > N and λ τ,1 − λ τ,N < p − N for any τ ∈ I F. Under this assumption, it follows from the crystallinity of ρ π at places of S p that for any w ∈ S p , ρ π | G Fw and ρ π | G Fw are Fontaine-Laffaille. (ORD π) F contains an imaginary quadratic field in which p splits, π is unramified and ordinary at all places v ∈ S p. Recall that (ORD π) implies that ρ π | G Fv takes values in a Borel of GL N (O). More precisely, there exists g v ∈ GL N (O) such that g v •ρ π | G Fv •g −1 v is upper triangular. Let B = T N be the Levi decomposition of the Borel B of upper triangular matrices with T the subgroup of diagonal matrices. We denote by χ v : G Fv → T (O) the homomorphism g v • ρ π | G Fv • g −1 v modulo N (O). We put χ v = diag(χ v,i) i=1,...,N and we write χ v,i for the reduction of χ v,i modulo ̟. With these notations, we consider the so-called pdistinguished condition (DIST) At each v ∈ S p , the characters χ v,i (i = 1,. .. , N) are mutually distinct.