In the local, characteristic 0, non-Archimedean case, we consider distributions on GL(n + 1) which are invariant under adjoint action of GL(n). We prove that such by transposition. This implies multiplicity at most one for restrictions from to Similar theorems obtained orthogonal or unitary groups.

In this paper we extend the notions of Schwartz functions, tempered functions and generalized to Nash (i.e. smooth semi-algebraic) manifolds. We reprove for case classically known properties on $R^n$ build some additional tools which are important in representation theory.

In the first part of paper we generalize a descent technique due to Harish-Chandra case reductive group acting on smooth affine variety both defined over an arbitrary local field F characteristic zero. Our main tool is Luna Slice Theorem. second apply this symmetric pairs. particular prove that pairs (GL(n+k,F), GL(n,F) x GL(k,F)) and (GL(n,E), GL(n,F)) are Gelfand for any its quadratic extension E. non-Archimedean case, result was proven earlier by Jacquet Rallis Flicker. We also...

In the local, characteristic 0, non-Archimedean case, we consider distributions on GL.n C 1/ which are invariant under adjoint action of GL.n/.We prove that such by transposition.This implies multiplicity at most one for restrictions from to GL.n/.Similar theorems obtained orthogonal or unitary groups.

We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction epimorphisms from model corresponding to a nilpotent orbit any same orbit, certain bigger orbits. also give choice-free definitions models. Finally, we explain how our methods imply analogous results Whittaker–Fourier coefficients automorphic representations. For $\text{GL}_{n}(\mathbb{F})$ this implies that...

Let F be an arbitrary local field. Consider the standard embedding of GL(n,F) into GL(n+1,F) and two-sided action \times on GL(n+1,F). In this paper we show that any GL(n,F)-invariant distribution is invariant with respect to transposition. We implies pair (GL(n+1,F),GL(n,F)) a Gelfand pair. Namely, for irreducible admissible representation $(π,E)$ (GL(n+1,F), $$dimHom_{GL(n,F)}(E,\cc) \leq 1.$$ For proof in archimedean case develop several new tools study distributions smooth manifolds.

Extending results of Kazhdan to the relative case, we relate harmonic analysis over some spherical spaces G(F)/H(F), where F is a field positive characteristic, G(E)/H(E), E suitably chosen characteristic 0. One Ingredients proof condition for finite generation modules Hecke algebra. We apply our show that pair (GL_{n+1},GL_n) strong Gelfand all local fields, and (GL_{n+k},GL_n x GL_k) fields odd characteristic.

We establish the existence of a transfer, which is compatible with Kloosterman integrals, between Schwartz functions on ${\rm GL}_n(\Bbb{R})$ and variety non-degenerate Hermitian forms. Namely, we consider an integral function along orbits two sided action groups upper lower unipotent matrices twisted by character. This gives smooth torus. prove that space all obtained in such way coincides constructed analogously when replaced hermitian also obtain similar results for...

In this paper we establish a connection between the associated variety of representation and existence certain {\it degenerate} Whittaker functionals, for both smooth $K$-finite vectors, all quasi-split real reductive groups, thereby generalizing results Kostant, Matumoto others.

We prove that any relative character (a.k.a. spherical character) of admissible representation a real reductive group with respect to pair subgroups is holonomic distribution on the group. This implies restriction an open dense subset given by analytic function. The proof based argument from algebraic geometry and thus also analogous results in p-adic case. As application, we give short some Kobayashi-Oshima Kroetz-Schlichtkrull boundedness finiteness multiplicities irreducible...

Let F be either ℝ or a finite extension of ℚ p , and let G central the group F-points reductive defined over F. Also π smooth representation (Fréchet moderate growth if F=ℝ). For each nilpotent orbit 𝒪 we consider certain Whittaker quotient π. We define support WS(π) to set maximal among those for which ≠0.

Abstract In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G an adelic split simply-laced group. Let $\pi $ be minimal or next-to-minimal representation . We prove that any $\eta \in \pi is completely determined by its Whittaker with respect to (possibly degenerate) characters the unipotent radical fixed Borel subgroup, analogously Piatetski-Shapiro–Shalika formula for cusp $\operatorname {GL}_n$ also derive explicit formulas expressing form, as well all...

Motivated by the theory of hypergeometric orthogonal polynomials, we consider quasi-orthogonal polynomial families, i.e. those that are with respect to a non-degenerate bilinear form defined linear functional, and in which ratio successive coefficients is given rational function $f(u,s)$ $u$. We call this family Jacobi type. Our main result there precisely five families These classical Jacobi, Laguerre Bessel two more one parameter $E_n^{(c)},F_n^{(c)}$. The last can be expressed through...

The goal of this paper is to describe the $\alpha$-cosine transform on functions a Grassmannian $i$-planes in an $n$-dimensional real vector space. analytic terms as explicitly possible. We show that for all but finitely many complex $\alpha$ composition $(\alpha+2)$-cosine with written (though complicated) O(n)-invariant differential operator. For exceptional values except one we interpret either Radon or two transforms. Explicit interpretation corresponding last remaining value $\alpha$,...

In the first part of paper we generalize a descent technique due to Harish-Chandra case reductive group acting on smooth affine variety both defined over arbitrary local field F characteristic zero. Our main tool is Luna slice theorem. second apply this symmetric pairs. particular prove that pair (GL(n,C),GL(n,R)) Gelfand pair. We also any conjugation invariant distribution GL(n,F) with respect transposition. For non-archimedean later classical theorem and Kazhdan. use techniques developed...

Let F be either ‫ޒ‬ or ‫.ރ‬Let (π, V ) an irreducible admissible smooth Fréchet representation of GL 2n (F).A Shalika functional φ : → ‫ރ‬ is a continuous linear such that for any g ∈ n (F), A Mat n×n (F) and v we haveIn this paper prove the space functionals on at most one-dimensional.For nonarchimedean (of characteristic zero) theorem was proved by Jacquet Rallis.