Alexander Premet
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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 $B_n~(n\ge 3)$, $D_n~(n\ge 4)$, $E_6, E_7, E_8$, $F_4$.
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…
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.
Gelfand-Kirillov dimensions and annihilator varieties of highest weight modules of exceptional Lie algebras
本文提出高效算法计算例外李代数单最高权模的Gelfand-Kirillov维数，并借助Sommers-Achar对偶确定其零化子簇。
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 $\mathfrak{g}$ 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 $L$.
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 …
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…
Irreducible representations of $\textrm{GL}_n(\mathbb{C})$ of minimal Gelfand-Kirillov dimension
本文通过Bernstein次数和Goldie秩多项式，比较了$\textrm{GL}_n(\mathbb{C})$最小Gelfand-Kirillov维数的不可约表示与极大抛物子群诱导表示，并给出相干族基的转移矩阵。
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 …
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.
Gelfand-Kirillov bound for $p$-adic Banach representations with infinitesimal character for $\text{GL}_2$ and quaternion units
本文研究了$p$-adic Banach表示的Gelfand-Kirillov界，针对$\text{GL}_2$和四元单位群，证明了具有微小特征的表示维度上限。
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…
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.
Dimension growth and Gelfand-Kirillov dimension of representations of quantum groups
本文研究了量子群表示的维度增长与Gelfand-Kirillov维度，揭示了非整数权表示对这两个不变量的影响，并确定了不同李型的最小非零值，进一步区分了量子与经典情况。
Gelfand--Kirillov dimension and mod $p$ cohomology for inner forms of $\mathrm{GL}_2$
本文计算了内形式 $D^\times$ 的模 $p$ 上同调中海德克特征空间的格尔福恩-基里洛夫维数，并在 $D$ 在 $p$ 处为除环时给出了简化证明。
Gelfand--Kirillov dimensions of highest weight modules for basic classical Lie superalgebras
本文开发了一种组合算法，用于计算基本经典李超代数的单最高权模的GK维数，并给出了类型sl(m|n)和osp(2|2n)的显式公式，证明GK维数完全由李超代数的偶部决定。
Computation of Gelfand-Kirillov dimension for $B$-type structures
本文计算了$B$型李群Spin和SO的正则函数量子化代数及其齐性空间上模的Gelfand-Kirillov维数，证明其分别等于Weyl字的长度和齐性空间的实流形维数。
Gelfand-Kirillov dimension for mod $p$ representations of $p$-adic unitary groups of rank 2
本文研究秩2 p进酉群的模p表示的Gelfand-Kirillov维数，在半单且足够一般的模L参数条件下，结合已有结果证明了相关容许光滑表示的Gelfand-Kirillov维数等于对应局部域在Q_p上的次数。
Gelfand-Kirillov dimension and mod p cohomology for GL2
本文研究全实域上四元数代数情形下 GL2 的模 p Galois 表示，证明相应 Hecke 特征空间中许多可容许光滑表示具有 Gelfand-Kirillov 维数 [F_v:Q]。
Irreducible non-holonomic modules for rational Cherednik algebras
该研究针对特征零代数闭域上的Weyl代数不变微分算子环，构造出特定维数的不可约非完整模，利用Morita等价推导有理Cherednik代数的对应结果，还证明Weyl代数在指定维数区间内存在所有维数的不可约模。
We study the endomorphism algebras of a modular Gelfand-Graev representation of a finite reductive group by investigating modular properties of homomorphisms constructed by Curtis and Curtis-Shoji.
Gelfand-Tsetlin modules: canonicity and calculations
本文介绍$\mathfrak{gl}_n$的Gelfand-Tsetlin模与图解KLRW代数的联系，证明单Gelfand-Tsetlin模的典范性，利用修改的Leclerc算法计算权空间维数，构造Verma模基，并计算秩3和4的模重数与Gelfand-Kirillov维数。
Corrigendum to "Modular Gelfand pairs and multiplicity-free representations"
本文指出“模Gelfand对与无重表示”中部分无重性结果过于宽泛，给出反例和部分修正。
Modular Gelfand pairs and multiplicity-free representations
本文推广 Gelfand 判据，证明任意特征代数闭域上 Gelfand 对与无重数表示定理，并用于 Whittaker 模型和三线性型唯一性。
A condition on an affine central subalgebra $Z$ of a noetherian algebra $A$ of finite Gelfand-Kirillov dimension, which we call here \emph{unruffledness}, is shown to be equivalent in some circumstances to the flatness of $A$ as a $Z$-module. Unruffledness was studied by Borho and Joseph in work on enveloping algebras of complex semisimple Lie algebras, and we discuss applications of our result to enveloping algebras, as well as beginning the study of this condition for more general algebras.
The main aim of this paper is to construct irreducible $(\mathfrak{g},K)$-modules of $\mathrm{O}(p,q)$ corresponding to the finite-dimensional representation of $\mathfrak{sl}_2$ of dimension $m+1$ under the Howe duality, to find the $K$-type formula, the Gelfand-Kirillov dimension and the Bernstein degree of them, where $m$ is a non-negative integer. The $K$-type formula for $m=0$ shows that it is nothing but the $(\mathfrak{g},K)$-module of the minimal representation of $\mathrm{O}(p,q)$. One finds that the Gelfand-Kirillov dimension is equal to $p+q-3$ not only for $m=0$ but for any $m$ sat…
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.
Intuitively, the filter dimension of an algebra or a module measures how `close' standard filtrations of the algebra or the module are. In particular, for a simple algebra it also measures the growth of how `fast' one can prove that the algebra is simple. The filter dimension appears naturally when one wants to generalize the Bernstein's inequality for the Weyl algebras to the class of simple finitely generated algebras. This paper is a review of the author's results on the filter dimension and its connections with the Gelfand-Kirillov dimension, the Krull dimension, the representation theory …
We establish the absence of zero divisors in the reduction algebra of a Lie algebra g with respect to its reductive Lie sub-algebra k. The class of reduction algebras include the Lie algebras (they arise when k is trivial) and the Gelfand--Kirillov conjecture extends naturally to the reduction algebras. We formulate the conjecture for the diagonal reduction algebras of sl type and verify it on a simplest example.
We study the modular representations of finite groups of Lie type arising in the cohomology of certain quotients of Deligne-Lusztig varieties associated with Coxeter elements. These quotients are related to Gelfand-Graev representations and we present a conjecture on the Deligne-Lusztig restriction of Gelfand-Graev representations. We prove the conjecture for restriction to a Coxeter torus. We deduce a proof of Broué's conjecture on equivalences of derived categories arising from Deligne-Lusztig varieties, for a split group of type $A\_n$ and a Coxeter element. Our study is based on Lusztig's …
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…
We define a class of quantum linear Galois algebras which include the universal enveloping algebra Uq(gln), the quantum Heisenberg Lie algebra and other quantum orthogonal Gelfand-Zetlin algebras of type A, the subalgebras of G-invariants of the quantum affine space, quantum torus for G = G(m, p, n), and of the quantum Weyl algebra for G = Sn. We show that all quantum linear Galois algebras satisfy the quantum Gelfand-Kirillov conjecture. Moreover, it is shown that the the subalgebras of invariants of the quantum affine space and of quantum torus for the reflection groups and of the quantum We…
On the annihilator variety of a highest weight Harish-Chandra module
本研究针对Hermitian型李群的最高权Harish-Chandra模，利用组合算法刻画其零化子簇，并证明可酉化情形下该模的Gelfand-Kirillov维数仅由最高根相关的参数值决定。
We prove a long-standing conjecture of Geck which predicts that cuspidal unipotent characters remain irreducible after $\ell$-reduction. To this end, we construct a progenerator for the category of representations of a finite reductive group coming from generalised Gelfand--Graev representations. This is achieved by showing that cuspidal representations appear in the head of generalised Gelfand--Graev representations attached to cuspidal unipotent classes, as defined and studied in \cite{GM96}.
Brown and Goodearl stated a conjecture that provides an explicit description of the topology of the spectra of quantum algebras. The conjecture takes on a more explicit form if there exist separating Ore sets for all incident pairs of torus invariant prime ideals of the given algebra. We prove that this is the case for the two largest classes of algebras of finite Gelfand-Kirillov dimension that fit the setting of the conjecture: the quantized coordinate rings of all simple algebraic groups and the quantum Schubert cell algebras for all symmetrizable Kac-Moody algebras.
Simple Iwasawa modules with large canonical dimension of quaternion algebra over $\mathbb{Q}_p$
构造四元数代数上大典范维数的绝对不可约Banach表示，作为Dospinescu-Schraen猜想的反例。
We provide Harish-Chandra type formulas for the multivariate Bessel functions and Heckman-Opdam hypergeometric functions as representation-valued integrals over dressing orbits. Our expression is the quasi-classical limit of the realization of Macdonald polynomials as traces of intertwiners of quantum groups given by Etingof-Kirillov Jr. Integration over the Liouville tori of the Gelfand-Tsetlin integrable system and adjunction for higher Calogero-Moser Hamiltonians recovers and gives a new proof of the integral realization over Gelfand-Tsetlin polytopes which appeared in the recent work of Bo…
Reducibility of Scalar Generalized Verma Modules of Minimal Parabolic Type II
本文研究了例外李代数中最小抛物子代数相关标量广义韦尔玛模的可约性，通过计算盖尔范-基里洛夫维数确定其第一个对角可约点。
For the modular lattice D^4 = {1+1+1+1} associated with the extended Dynkin diagram \tilde{D}_4 (and also for D^r, where r > 4), Gelfand and Ponomarev introduced the notion of admissible and perfect lattice elements and classified them. In this work, we classify the admissible and perfect elements in the modular lattice D^{2,2,2} = {2+2+2} associated with the extended Dynkin diagram \tilde{E}_6. Gelfand and Ponomarev constructed admissible elements for D^r recurrently in the length of multi-indices, which they called admissible sequences. Here we suggest a direct method for creating admissible…
This paper provides results on the modular representation theory of the supergroup $GL(m|n).$ Working over a field of arbitrary characteristic, we prove that the explicit combinatorics of certain crystal graphs describe the representation theory of a modular analogue of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$. In particular, we obtain a linkage principle and describe the effect of certain translation functors on irreducible supermodules. Furthermore, our approach accounts for the fact that $GL(m|n)$ has non-conjugate Borel subgroups and we show how Serganova's odd reflections give…
Posets of finite GK-dimensional graded pre-Nichols algebras of diagonal type
本文分类了具有有限 Gelfand-Kirillov 维数的对角型分次 pre-Nichols 代数，通过正根子集的偏序集同构进行刻画，推广了小量子群的中心扩张，并分类了相关幂么代数群函数代数的分次商。
Let $G_{\mathbb{R}}$ be a simple real linear Lie group with maximal compact subgroup $K_{\mathbb{R}}$ and assume that ${\rm rank}(G_\mathbb{R})={\rm rank}(K_\mathbb{R})$. In \cite{MPVZ} we proved that for any representation $X$ of Gelfand-Kirillov dimension $\frac{1}{2}\dim(G_{\mathbb{R}}/K_{\mathbb{R}})$, the polynomial on the dual of a compact Cartan subalgebra given by the dimension of the Dirac index of members of the coherent family containing $X$ is a linear combination, with integer coefficients, of the multiplicities of the irreducible components occurring in the associated cycle. In t…
Let $G_{\mathbb{R}}$ be a simple real linear Lie group with maximal compact subgroup $K_{\mathbb{R}}$ and assume that ${\rm rank}(G_\mathbb{R})={\rm rank}(K_\mathbb{R})$. For any representation $X$ of Gelfand-Kirillov dimension $\frac{1}{2} {\rm dim}(G_{\mathbb{R}}/K_{\mathbb{R}})$, we consider the polynomial on the dual of a compact Cartan subalgebra given by the dimension of the Dirac index of members of the coherent family containing $X$. Under a technical condition involving the Springer correspondence, we establish an explicit relationship between this polynomial and the multiplicities of…
The goal of this paper is to generalize several basic results from the theory of $\cal{D}$-modules to the representation theory of rational Cherednik algebras. We relate characterizations of holonomic modules in terms of singular support and Gelfand-Kirillov dimension. We study pullback, pushforward, and dual on the derived category of (holonomic) Cherednik modules for certain classes of maps between varieties. We prove, in the case of generic parameters for the rational Cherednik algebra, that pushforward with respect to an open affine inclusion preserves holonomicity.
Finite GK-dimensional Nichols algebras over the quaternion group $Q_8$
本文研究四元数群 $Q_8$ 上的 Nichols 代数，分类其全部不可约 Yetter-Drinfeld 模，并判定相应 Nichols 代数何时具有有限 GK 维数或有限维，以推进有限 GK 维 Hopf 代数的分类。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。