We prove that an admissible $p$-adic Banach representation of $\text{GL}_2K$ whose locally analytic vectors have an infinitesimal character has Gelfand-Kirillov dimension $\leq[K\colon\mathbf Q_p]$, where $p>2$ and $K$ is a $p$-adic field. We also prove this for the group of units of the quaternions over $K$ replacing $\text{GL}_2K$. In the process, we make some observations in the theory of $p$-adic Banach representations that might be of independent interest.
We find an exact formula of Gelfand-Kirillov dimensions for the infinite-dimensional explicit irreducible sl(n, F)-modules that appeared in the Z^2-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. Three infinite subfamilies of these modules have the minimal Gelfand-Kirillov dimension. They contain weight modules with unbounded weight multiplicities and completely pointed modules.
We calculate the asymptotic behavior of the dimension of the fixed vectors of $π$ with respect to compact open subgroups $1+ M_n(\mathfrak{p}^N)\subset\mathrm{GL}_n(F)$ for $π$ an admissible representation of $\mathrm{GL}_n(F)$, and $F$ a nonarchimedean local field. Such dimensions can be calculated by germs of the character of $π$. We also make some observations on how those dimensions behave under instances of Langlands functoriality, such as the Jacquet-Langlands correspondence and cyclic base change, where relations between characters are known.
Let g be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero. We show that if the Gelfand-Kirillov conjecture holds for g, then g has type A_n, C_n or G_2.
Let $\mathfrak{g}$ be a classial Lie algebra and $\mathfrak{p}$ be a maximal parabolic subalgebra. Let $M$ be a generalized Verma module induced from a one dimensional representation of $\mathfrak{p}$. Such $M$ is called a scalar type generalized Verma module. Its simple quotient $L$ is a highest weight moudle. In this paper, we will determine the reducibility of such scalar type generalized Verma modules by computing the Gelfand-Kirillov dimension of $L$.
Consider the W-algebra $W$ attached to the smallest nilpotent orbit in a simple Lie algebra $\frak g$ over an algebraically closed field of characteristic 0. We show that if an analogue of the Gelfand-Kirillov conjecture holds for such a W-algebra then it holds for the universal enveloping algebra $\mathrm U(\frak g)$. This together with a result of A. Premet implies that the analogue of the Gelfand-Kirillov conjecture fails for some $W$-algebras attached to some nilpotent orbits in Lie algebras of types $B_n~(n\ge 3)$, $D_n~(n\ge 4)$, $E_6, E_7, E_8$, $F_4$.
In this paper, we compute the Gelfand-Kirillov dimensions for the infinite-dimensional explicit irreducible $\mathfrak{o}(n, \mathbb{C})$-modules and $\mathfrak{sp}(2n, \mathbb{C})$-modules that appeared in the $\mathbb{Z}$-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. We also found that some of the modules have the secondly minimal GK-dimension, and some of them have the larger GK-dimension than the maximal GK-dimension of unitary highest-weight modules.
Recently, Anno, Bezrukavnikov and Mirkovic have introduced the notion of a "real variation of stability conditions" (which is related to Bridgeland's stability conditions), and construct an example using categories of coherent sheaves on Springer fibers. Here we construct another example of representation theoretic significance, by studying certain sub-quotients of category O with a fixed Gelfand-Kirillov dimension. We use the braid group action on the derived category of category O, and certain leading coefficient polynomials coming from translation functors. Consequently, we use this to expl…
Let $ (G,K) $ be an irreducible Hermitian symmetric pair of non-compact type with $G=SU(p,q)$, and let $ λ$ be an integral weight such that the simple highest weight module $ L(λ) $ is a Harish-Chandra $ (\mathfrak{g},K) $-module. We give a combinatoric algorithm for the Gelfand-Kirillov dimension of $ L(λ) $. This enables us to prove that the Gelfand-Kirillov dimension of $ L(λ) $ decreases as the integer $ \langleλ+ρ,β^\vee\rangle $ increases, where $ρ$ is the half sum of positive roots and $β$ is the maximal noncompact root. As a byproduct, we obtain a description on the associated variety …
We address two problems regarding the structure and representation theory of finite W-algebras associated with the general linear Lie algebras. Finite W-algebras can be defined either via the Whittaker model of Kostant or, equivalently, by the quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of the finite W-algebras. The second main result is a parametrization of finite families of irreducible Harish-Chandra modules by the characters of the Gelfand-Tsetlin subalgebra. As a corollary, we obtain a complete classif…
In this paper, we present a uniform formula of Lusztig's $ \mathbf{a}$-functions on classical Weyl groups. Then we obtain an efficient algorithm for the Gelfand-Kirillov dimensions of simple highest weight modules of classical Lie algebras, whose highest weight is not necessarily regular or integral. To deal with type $ D $, we prove an interesting property about domino tableaux by introducing an invariant, called the hollow tableau. As an application, the associated varieties of all the simple highest weight Harish-Chandra modules are explicitly determined, including the exceptional cases.
It is shown that the q-difference Noether problem for all classical Weyl groups has a positive solution, simultaneously generalizing well known results on multisymmetric functions of Mattuck and Miyata in the case q=1, and q-deforming the noncommutative Noether problem for the symmetric group. It is also shown that the quantum Gelfand-Kirillov conjecture for gl_N (for a generic q) follows from the positive solution of the q-difference Noether problem for the Weyl group of type D_n. The proof is based on the theory of Galois rings developed by the first author and Ovsienko. From here we obtain …
We construct functorially a class of algebras using the formalism of double derivations. These algebras extend to higher dimensions Crawley-Boevey and Holland's construction of deformed preprojective algebras and encompass symplectic reflection algebras associated to wreath products. We use this construction to show that the quotient field of a symplectic reflection algebra is "rational", confirming a pair of conjectures of Etingof and Ginzburg.
Let $\mf g$ be the Witt algebra or the positive Witt algebra. It is well known that the enveloping algebra $U(\mf g )$ has intermediate growth and thus infinite Gelfand-Kirillov (GK-) dimension. We prove that the GK-dimension of $U(\mf g)$ is {\em just infinite} in the sense that any proper quotient of $U(\mf g)$ has polynomial growth. This proves a conjecture of Petukhov and the second named author for the positive Witt algebra. We also establish the corresponding results for quotients of the symmetric algebra $S(\mf g)$ by proper Poisson ideals. In fact, we prove more generally that any cent…
Macdonald polynomials are an important class of symmetric functions, with connections to many different fields. Etingof and Kirillov showed an intimate connection between these functions and representation theory: they proved that Macdonald polynomials arise as (suitably normalized) vector-valued characters of irreducible representations of quantum groups. In this paper, we provide a branching rule for these characters. The coefficients are expressed in terms of skew Macdonald polynomials with plethystic substitutions. We use our branching rule to give an expansion of the characters with respe…
Let $G=GL_{n}(\mathbb{C})$ and $1\neψ:\mathbb{C}\to\mathbb{C}^{\times}$ be an additive character. Let $U$ be the subgroup of upper triangular unipotent matrices in $G$. Denote by $θ$ the character $θ:U\to\mathbb{C}$ given by \[ θ(u):=ψ(u_{1,2}+u_{2,3}+...+u_{n-1,n}). \] Let $P$ be the mirabolic subgroup of $G$ consisting of all matrices in $G$ with the last row equal to $(0,0,...,0,1)$. We prove that if $π$ is an irreducible generic representation of $GL_{n}(\mathbb{C})$ and $\mathcal{W}(π,ψ)$ is its Whittaker model, then the space $\{f|_{P}:P\to \mathbb{C}:\, f\in \mathcal{W}(π,ψ)\}$ contains…
The unitary principal series representations of $G=GL(n,\mathbb{C})$ induced from a character of the maximal parabolic subgroup $P=(GL(1,\mathbb{C})\times GL(n-1,\mathbb{C}))\ltimes\mathbb{C}^{n-1}$ attain the minimal Gelfand--Kirillov dimension among all infinite-dimensional unitary representations of $G$. We find the explicit branching laws for the restriction of these representations to symmetric subgroups of $G$.
We study finite dimensional representations over some Noetherian algebras over a field of characteristic zero. More precisely, we give necessary and sufficient conditions for the category of locally finite dimensional representations to be closed under taking injective hulls and extend results known for group rings and enveloping algebras to Ore extensions, Hopf crossed products and affine Hopf algebras of low Gelfand-Kirillov dimension.
In this paper we review and streamline some results of Kirillov, Olshanski and Pickrell on unitary representations of the unitary group $\U(\cH)$ of a real, complex or quaternionic separable Hilbert space and the subgroup $\U_\infty(\cH)$, consisting of those unitary operators $g$ for which $g - \1$ is compact. The Kirillov--Olshanski theorem on the continuous unitary representations of the identity component $\U_\infty(\cH)_0$ asserts that they are direct sums of irreducible ones which can be realized in finite tensor products of a suitable complex Hilbert space. This is proved and generalize…
