Explicit Hilbert spaces for certain unipotent representations III

On R-duals of type III in Hilbert spaces

Following work by Casazza, Kutyniok and Lammers, and extensions by Stoeva and Christensen, we provide some novel characterizations of R-dual sequences of type III in Hilbert spaces. We systematically extend the construction procedure by basing it on a choice of an antiunitary involution. For certain classes of R-duals of type III, we derive a representation of the associated frame operator in terms of spectral measures.

有限域上Howe对偶III：完整计算与Gurevich-Howe猜想

Howe duality over finite fields III: Full computation and the Gurevich-Howe conjectures

本文研究有限域上Howe对偶的完整计算，探讨对称群和正交群的振荡表示限制，并证明Gurevich-Howe猜想。

秩一正交群酉对偶的显式 Hilbert 空间及其应用

Explicit Hilbert spaces for the unitary dual of rank one orthogonal groups and applications

本文在去原点欧氏空间的显式向量值 $L^2$ Hilbert 空间中实现 $\mathrm{SO}_0(n+1,1)$ 的全部不可约酉表示，并借助 Knapp–Stein 算子刻画 Whittaker 向量及抛物子群分解。

Moduli of affine schemes with reductive group action

For a connected reductive group G and a finite-dimensional G-module V, we study the invariant Hilbert scheme that parameterizes closed G-stable subschemes of V affording a fixed, multiplicity-finite representation of G in their coordinate ring. We construct an action on this invariant Hilbert scheme of a maximal torus T of G, together with an open T-stable subscheme admitting a good quotient. The fibers of the quotient map classify affine G-schemes having a prescribed categorical quotient by a maximal unipotent subgroup of G. We show that V contains only finitely many multiplicity-free G-subva…

Notes on quantization of symplectic vector spaces over finite fields

In these notes we construct a quantization functor, associating an Hilbert space H(V) to a finite dimensional symplectic vector space V over a finite field F_q. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp(V). The main technical result is a proof of a stronger form of the Stone-von Neumann theorem for the Heisenberg group over F_q. Our result answers, for the case of the Heisenberg group, a question of Kazhdan about the possible existence of a canonical Hilbert space attached to a coadjoint orbit of a general unipotent group over F_q.

Analysis on the minimal representation of O(p,q) -- III. ultrahyperbolic equations on R^{p-1,q-1}

For the group O(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of the q = 2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The group O(p,q) acts as the Moebius group of conformal transformations on R^{p-1, q-1}, and preserves a space of solutions of the ultrahyperbolic Laplace equation on R^{p-1, q-1}. We construct in an intrinsic and natural way a Hilbert space of ultrahyperbolic solutions so that O(p,q) becomes a continuous irre…

The Basic Representation of the Current Group O(n,1)^X in the L^2 space over the generalized Lebesgue Measure

We give the realization of the representation of the current group O(n,1)^X where X is a manifold, in the Hilbert space of L^2(F,ν) of functionals on the the space F of the generalized functions on the manifold X which are square integrable over measure νwhich is related to a distinguish Levy process with values in R^{n-1} which generalized one dimensional gamma process. Unipotent subgroup of the group O(n,1)^X acts as the group of multiplicators. Measure νis sigma-finite and invariant under the action current group O(n-1)^X. Ther case of n=2 (SL(2,R^X)) was considered before in the series of …

反射方程代数的表示论 III：不可约表示的分类

Representation theory of the reflection equation algebra III: Classification of irreducible representations

本文发展最高权理论，分类反射方程代数（由 $GL(N,\mathbb{C})$ 的 $q$-变形 $R$-矩阵构造）的不可约有界 $*$-表示。

Infinite Dimensional Multiplicity Free Spaces III: Matrix Coefficients and Regular Functions

In earlier papers we studied direct limits $(G,K) = \varinjlim (G_n,K_n)$ of two types of Gelfand pairs. The first type was that in which the $G_n/K_n$ are compact Riemannian symmetric spaces. The second type was that in which $G_n = N_n\rtimes K_n$ with $N_n$ nilpotent, in other words pairs $(G_n,K_n)$ for which $G_n/K_n$ is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius--Schur Orthogonality Relations to define isometric injections $ζ_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)$ for $m \geqq n$ and prove that the left regular representation of $G$ on …

An analogue of Hilbert's Theorem 90 for infinite symmetric groups

Let $K$ be a field and $G$ be a group of its automorphisms. If $G$ is precompact then $K$ is a generator of the category of smooth (i.e. with open stabilizers) $K$-semilinear representations of $G$. There are non-semisimple smooth semilinear representations of $G$ over $K$ if $G$ is not precompact. In this note the smooth semilinear representations of the group $G$ of all permutations of an infinite set $S$ are studied. Let $k$ be a field and $k(S)$ be the field freely generated over $k$ by the set $S$ (endowed with the natural $G$-action). One of principal results describes the Gabriel spectr…

Hilbert-Lie群的协变射影表示

Covariant projective representations of Hilbert-Lie groups

本文研究Hilbert-Lie群的酉表示，刻画有界表示的离散可分解性，并利用协变性发展微扰理论，最后扩展到射影表示并显式确定中心扩张。