Correspondances de Howe sur un corps p-adique

Over the past several years, operator algebraists have become increasingly interested in the problem of calculating the K-theory of group C∗-algebras. The focal point of research in this area ist he BaumConnesConjecture[BCH],which proposes a description of K-theory for the C∗-algebra of a group in terms of homology and the representation theory of compact subgroups. Although the main applications of the Baum-Connes Conjecture are to issues in geometry and topology, the conjecture also appears to be of interest from the point of view of harmonic analysis. Whereas for applications to topology one is concerned with discrete groups G (arising as the fundamental groups of manifolds), the conjecture’s links with harmonic analysis appear to be the strongest for reductive Lie groups and p-adic groups. The purpose of these notes is to convey to a reasonably broad audience some byproducts of the authors’ research into the C∗-algebra K-theory of the p-adic group GL(N), which culminated in a proof of the Baum-Connes Conjecture in this case [BHP2]. Along the way to the proof a number of interesting issues came to light which we feel deserve some exposure, even though our understanding of them is far from complete, and is indeed mostly very tentative. Much of what follows is focused on what we call here chamber homology, which is a type of equivariant homology associated to the action of a reductive p-adic group on its Bruhat-Tits affine building. The problem of computing chamber homology can be approached from a number of different directions. An especially interesting problem is to reconcile chamber homology with the Bernstein decomposition for representations of reductive p-adic groups [Be, BD]. This appears to be a far from trivial matter, even in comparatively simple cases. In Section 5 we formulate two very general conjectures which give a broad description of a Bernstein decomposition in chamber homology (perhaps we should call our conjectures questions, since the evidence we have gathered in their favor is not overwhelming).

Weil's representation is a basic object in representation theory which plays a crucial role in many places: construction of unitary irreducible representations in the frame of the orbit method, Howe correspondence, Theta series, … The decomposition in irreducibles of the restriction of Weil's representation to maximal compact subgroups or anisotropic tori of the metaplectic group is thus an important information in representation theory. Except for $SL(2)$, this was not known in the p-adic case. In this article, we prove that the restriction of the Weil representation over a p-adic field, $p\neq2$, to maximal compact subgroups is multiplicity free and give an explicit description of the irreducibles occurring. In another paper, using our results, we describe the decomposition of the restriction of the Weil representation to maximal elliptic tori.

The object of this paper is to derive some explicit formulae concerning the Weil representation that allow us to define this projective representation in a unique manner for each choice of symplectic basis.Let F be a self-dual locally compact field of char φ 2 and X a symplectic vector space over F. Let V, V* be two transversal Lagrangian subspaces.Then a classical construction due to Shale-Segal-Weil gives a projective representation of the symplectic group Sp(JSΓ) in the Schwartz-space of V.The operators ξ(σ) corresponding to each σ e Sp(X) are determined uniquely only up to a scalar multiple.The starting point of this paper is an explicit integral formula for these operators ξ(σ), valid for all σ e Sp(X).In fact (see Lemma 3.2) we have for each σ e Sp(ΛΓ) ξ(σ)φ :x-+ f σ (x, x*)φ(xa + x*y) dμ σ JV*/kevγwhere μ σ is a Haar measure on F*/kery, and f σ is the character of second degree on X, associated to σ.Here a -[" £] is the matrix representation of σ in the decomposition X = V + V*.This formula is known and already present in Weil's paper when γ = 0 or when γ is an isomorphism.The extension of its validity for all σ enables us to show that it is possible to define the projective representation in a unique way for each choice of symplectic basis.Let β\,... , e n , e\, ... , e* be a symplectic basis of X such that e\, ... , e n {e\,... , e*) is a basis of V (V*).Let W be the finite subgroup of Sp(X) consisting of all σ such that {ei, e*}σ c {±ei, ±ef} for each /.Then one has the well-known Bruhat decomposition Sp(X) = PWP, where P is the stabilizer of F*.Then it is shown that it is possible to make consistent choices of the Haar measures μ σ so that (1) ζ{p\σp2) = ζ(Pι)ζ{^)ζ(p2) f°Γ all Pup 2 e P and (2) ξ( σι σ 2 ) = ξ(σ ι )ζ(σ 2 ) for all σ Ϊ9 σ 2 e W. Moreover all such are determined.Among these there is one choice σ -• r{σ) called the standard model which in addition satisfies nonnegativity properties similar to those of the Fourier Transform.All

Introduction. In the investigation of a non-degenerate quadratic form f (x) with coefficients in a local or global field K, one associates two groups with f (x), the orthogonal and metaplectic groups. If K is, e.g., the field R of real numbers, the second group is the unitary group generated by Ut = exp(it -f(x)) and its conjugate V, under the Fourier transformation for all t in R; and it is locally isomorphic to SL2(R). By using a similar definition one can associate with any polynomial f (x) of higher degree a group. Such groups have appeared in connection with the Siegel-Weil formulas for the norm forms of certain simple Jordan algebras. On the other hand Kubota [4], [5] and Yamazaki [8] have associated for K = C and R metaplectic groups with where n need not be 2, in fact n 2 and n = 2k 2, respectively. In both cases they have used certain Bessel transformations instead of the Fourier transformation to get the metaplectic groups. In a recent paper [3] we have shown that the Lie algebra of a hypermetaplectic group over R is infinite dimensional (at least if the Fourier transformation is defined relative to the bilinear form associated with a definite quadratic form). In this paper we have restricted ourselves to the one-variable case and tried to obtain all metaplectic groups over K = R and C. More precisely we have started from an arbitrary polynomial f (x) of one variable x with coefficients in K = R or C, put U, = exp (it .f (x)) or exp(2iRe(tf(x)) and determined all cases where U, and another oneparameter subgroup V, of the unitary group of a certain L2-space generate a Lie group G locally isomorphic to SL2(K). In doing so we have assumed that the tangent vector or vectors of V, at t = 0 is a linear differential operator with polynomial coefficients, a condition satisfied in the known cases. The result is that f (x) simply becomes x, x2 under a suitable normalization and, accordingly, the differential operator becomes i-times

Let G denote Sp(n,R), U(p,q), or O∗(2n). The main aim of this article is to show that the canonical quantization of the moment map on a symplectic G-vector space (W,ω) naturally gives rise to the oscillator (or Segal–Shale–Weil) representation of g:=Lie(G)⊗C. More precisely, after taking a complex Lagrangian subspace V of the complexification of W, we assign an element of the Weyl algebra for V to 〈μ,X〉 for each X∈g, which we denote by 〈μˆ,X〉. Then we show that the map X↦i〈μˆ,X〉 gives a representation of g. With a suitable choice of V in each case, the representation coincides with the oscillator representation of g.

In this paper we study certain category of smooth modules for reductive p-adic groups analogous to the usual smooth complex representations but with the field of complex numbers replaced by a ℚ-algebra. We prove some fundamental results in these settings, and as an example we give a classification of admissible unramified irreducible representations using the reduction to the complex case.

The canonical measure on a reductive p-adic group is motivic

Extraspecial towers and weil representations

We investigate the ramification of embeddings of local lattices or quadratic forms over the ring of -adic numbers. It is proved that every primitive embedding decomposes uniquely into an orthogonal sum of minimal indecomposable embeddings, and all such embeddings are constructed for -elementary lattices. Ramification theory enables us to find the number of orbits of representations for forms and, in particular, for numbers by other quadratic forms over , and to calculate the local multipliers in the weight formula for representations of a form by a genus of quadratic forms.

The study of groups has been of interest to mathematicians since the 19th century. Although much is known about the structure of groups, many group theoretic problems remain unsolved. Representation theory allows us to employ linear algebra to solve such problems. The representation theory of linear groups over finite fields has been a particularly interesting topic. Studying these representations is of interest to mathematicians and other scientists as it relates to physics and modern number theory. In the 1960s Andre Weil introduced a method for finding a special unitary representation for symplectic groups over locally compact fields. This unitary representation is now referred to as the Weil representation. In 2010 Luis Gutierrez, Jose Pantoja and Jorge Soto-Andrade were able to generalize Weil’s method to a larger class of linear groups namely the ∗-analogue of Sl2. Originally, Weil constructed this unitary representation, decomposed it into irreducibles and, in this way, produced the irreducible complex representations of Sp(2n, k). Later, Shalika went in the other direction, first finding the irreducible representations and then computing their multiplicities in the Weil representation. We intend to follow Shalika’s method. In this thesis we look to explore the representation theory of Sl∗(2,A) where A is the direct sum of the upper and lower n × n block matrices in M(2n, k), k a finite field. We use Wigner and Mackey’s Method of Little Groups to construct these representations.

Let G be a reductive p-adic group and let Rep(G)s be a Bernstein block in the category of smooth complex G-representations. We investigate the structure of Rep(G)s, by analysing the algebra of G-endomorphisms of a progenerator Π of that category.We show that Rep(G)s is “almost” Morita equivalent with a (twisted) affine Hecke algebra. This statement is made precise in several ways, most importantly with a family of (twisted) graded algebras. It entails that, as far as finite length representations are concerned, Rep(G)s and EndG(Π)-Mod can be treated as the module category of a twisted affine Hecke algebra.We draw two major consequences. Firstly, we show that the equivalence of categories between Rep(G)s and EndG(Π)-Mod preserves temperedness of finite length representations. Secondly, we provide a classification of the irreducible representations in Rep(G)s, in terms of the complex torus and the finite group canonically associated to Rep(G)s. This proves a version of the ABPS conjecture and enables us to express the set of irreducible G-representations in terms of the supercuspidal representations of the Levi subgroups of G.Our methods are independent of the existence of types, and apply in complete generality.

For a reductive p-adic group G, we compute the supports of the Hecke algebras for the K-types for G lying in a certain frequently-occurring class. When G is classical, we compute the intertwining between any two such K-types.

Harmonic Analysis on Reductive p-adic Groups

Harmonic Analysis on Reductive p-adic Groups

Over a finitely generated field extension in m variables over a p-adic field, any quadratic form in more than 2m + 2 variables has a nontrivial zero. This bound is sharp. We extend this result to a wider class of fields. A key ingredient to our proofs is a recent result of Heath-Brown on systems of quadratic forms over p-adic fields.