Central morphisms and Cuspidal automorphic Representations

Journal of Algebra, 1989

Let G be a connected semisimple algebraic group over a field of non-zero characteristic p, with a maximal torus T. On G we have the Frobenius endomorphism F induced by the pth power map on the underlying field. The (scheme-theoretic) kernel of F is the first Frobenius kernel G,, and the inverse image of T under F is the "Frobenius subgroup" G, T. The purpose of this paper is to study certain questions in the (rational) representation theory of G, T. Let B be a Bore1 subgroup of G which contains T. In [6], Cline, Parshall, and Scott obtained a result which describes the functor "induction from B to G" as the composite of functors of the form "induction from B to a minimal parabolic subgroup P" over a sequence of minimal parabolics coming from a given reduced expression of the long word in the Weyl group. In Section 5, we obtain the analogous result for G, T. In Section 6, we develop a theory of intertwines for a certain class of G, T-modules. This gives a direct proof of an extended strong linkage principle for G, T, independent of the strong linkage principle for G. (Originally, Jantzen [17] proved that strong linkage for G, T, and more generally for G, T, n > 1, follows from the strong linkage principle for G. 1 Recently we have applied this method to give a new proof of an extended strong linkage principle for G; see [ 121. From the results of Section 5 we can write down a formula for the image of one of the intertwines in our class of modules. (This also follows from results of Jantzen [16, 171.) Composing the intertwines in question, relative to the class of modules corresponding to a given character of T, gives a map whose image is the irreducible G, T-module of that highest weight. In case the character is a restricted dominant character, the image coincides with the irreducible G-module of that highest weight, by a result

2022

Let p and l be distinct primes and let n be a positive integer. Let E be a finite Galois extension of degree l of a p-adic field F. Let π and ρ be two l-adic integral smooth cuspidal representations of GLn(E) and GLn(F) respectively such that π is obtained as base change of ρ. Then the Tate cohomology H 0 (π), as an l-modular representation of GLn(F), is well defined. In this article, we show that H 0 (π) is isomorphic to the Frobenius twist of the reduction mod-l of the representation ρ. We assume that l, p = n and l is banal for GL n−1 (F). SABYASACHI DHAR AND SANTOSH NADIMPALLI defined in the work of G.Moss and N.Matringe in [MM22]. However, the right notion of Base change over local Artinian F l-algebras is not clear to the authors and hence, we use the mild hypothesis that l is banal for GL n−1 (F). When F is a local function field, the above theorems follow from the work of T.Feng [Fen20]. T.Feng uses the constructions of Lafforgue and Genestier-Lafforgue [GL17]. N.Ronchetti also proved the above results for depth-zero cuspidal representations using compact induction model. Our methods are very different from the work of N.Ronchetti and the work of T.Feng. We rely on Rankin-Selberg integrals and lattices in Whittaker models. We do not require the explicit construction of cuspidal representations. We use the l-adic local Langlands correspondence and various properties of local ǫ and γ-factors both in l-adic and mod-l situations associated with the representations of the p-adic group and the Weil group. The machinery of local ǫ and γfactors of both l-adic and mod-l representations of GL n (F) is made available by the seminal works of D.Helm, G.Moss, N.Matringe and R.Kurinczuk (see [HM18], [Mos16], [KM21], [KM17]). We sketch the proof of the above Theorem (1.1). It is proved using Kirillov model and using some results of Vigneras on the lattice of integral functions in a Kirillov model being an invariant lattice. The discussion in this paragraph is valid for any positive integer n. Let ψ : F → Q × l be a non-trivial additive character. Let (π F , V) be a generic l-adic representation of GL n (F), in particular, V is a Q l-vector space. Let N n (F) be the group of unipotent upper triangular matrices in GL n (F). Let Θ : N n (F) → Q × l be a non-degenerate character corresponding to ψ and we let W (π F , ψ) to be the Whittaker model of π F. Similar notations for π E are followed. Let π E be the base change of π F. It is easy to note that (Lemma 2.3) W (π E , ψ) is stable under the action of Gal(E/F) on the space Ind GLn(E) Nn(E) Θ. Let π F be an integral generic l-adic representation of GL n (F), and let W 0 (π F , ψ) be the set of Z l-valued functions in W (π F , ψ). It follows from the work of Vigneras [Vig04, Theorem 2] that the subset W 0 (π F , ψ) is a GL n (F) invariant lattice. Let K(π F , ψ) be the Kirillov model of π F , and let K 0 (π F , ψ) be the set of Z l-valued functions in K(π F , ψ). Using the result [MM22, Corollary 4.3] we get that the restriction map from W 0 (π F , ψ) to K 0 (π F , ψ) is a bijection. Let π F and π E be cuspidal representations. It follows from the work of Treumann-Venkatesh that H(Gal(E/F), K 0 (π E , ψ)) is equal to K(r l (π F) (l) , ψ l), as representations of P n (F), where P n (F) is the mirabolic subgroup of GL n (F). Now, to prove the main theorem it is enough to check the compatibility of the action of π E (w) on the space H 0 (Gal(E/F), K 0 (π E , ψ)) with the action of r l (π F) (l) on K(r l (π F) (l) , ψ l). Such a verification, as is well known, involves arithmetic properties of the representations. We now assume the hypothesis of theorem (1.1). To complete the proof we show that it is enough to prove that

Japanese Journal of Mathematics, 2020

The functoriality conjecture of Langlands, as stated in [33], offers a general organizational scheme for all automorphic representations. This conjecture can be roughly stated as follows: if H and G are reductive groups defined over a global field k, ξ : L H → L G a homomorphism between their Langlands dual groups, then it is possible to attach to each automorphic representation π H = ⊗ v π H v of H a packet of automorphic representations π = ⊗ v π v of G. In the formula π = ⊗ v π v , local components π v are irreducible admissible representations of G(k v), k v being the completion of k at a place v ∈ |k|. The automorphic functoriality is expected to be of local nature in the sense that it should be compatible with the local functoriality principle that consists in a map from the set of packets of irreducible admissible representations of H(k v) to the set of packets of irreducible admissible representations of G(k v). Since unramified representations of H(k v) at an unramified nonarchimedean place v are parametrized by certain semi-simple conjugacy classes in L H, the local functoriality is required to be compatible with the transfer of conjugacy classes from L H to L G. A crucial part of the functoriality program, known as endoscopy, is now almost completely understood thanks to cumulative efforts of many mathematicians over the last 40 years, see [5]. Its main ingredients include in particular the construction of the general trace formula, mainly due to Arthur [4] and the proof of the transfer conjecture and the fundamental lemma, see [54] and [39]. The endoscopic case is concerned with the case where L H is essentially the centralizer of a semi-simple element in L G. In spite of this severe restriction, the endoscopic case is crucial in the general picture for it gives a sense to the concept of packet, both in local and automorphic settings, along with global multiplicity formulae [31]. The endoscopic program also produced the stable trace formula expected to be an essential tool for future works on the conjecture of functoriality. In [30], Langlands proposed a conjectural limiting form of the stable trace formula aiming at isolating the part of the automorphic spectrum of G that comes from a smaller group H by functoriality. These limiting forms of trace formulae are designed to have a spectral expansion involving logarithmic

Progress in Mathematics

Israel Journal of Mathematics, 2010

We define a new notion of cuspidality for representations of GL n over a finite quotient o k of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups G λ of torsion o-modules. When n is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of GL n (F). We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, we show that the construction of the representations theory of GL n (o k) for k ≥ 2 for all n is equivalent to the construction of the representations of all the groups G λ. A functional equation for zeta functions for representations of GL n (o k) is established for representations which are not contained in an infinitesimally induced representation. In the appendix, all cuspidal representations for GL 4 (o 2) are constructed. Not all these representations are strongly cuspidal.

Journal of the European Mathematical Society, 2000

Compositio Mathematica, 2004

We generalize a theorem of Tate and show that the second cohomology of the Weil group of a global or local field with coefficients in C * (or, more generally, with coefficients in the complex points of an algebraic torus over C) vanish, where the cohomology groups are defined using measurable cochains in the sense of Moore. We recover a theorem of Labesse stating that the admissible homomorphisms of a Weil group to the Langlands dual group of a reductive group can be lifted to an extension of the Langlands dual group by a torus.

2005

Geometric And Functional Analysis, 2004

We introduce a categorical framework for the study of representations of G(F), where G is a reductive group, and F is a 2-dimensional local field, i.e., F = K((t)), where K is a local field. Our main result says that the space of functions on G(F), which is an object of a suitable category of representations of G(F) with the respect to the action of G on itself by left translations, becomes a representation of a certain central extension of G(F), when we consider the action by right translations.