**1**

vote

**0**answers

140 views

### Derived category of representations

Suppose we are given an algebraic group $G$ (linearly reductive). Let $D^b(Repr(G))$ be the bounded derived category of finite dimensional algebraic representations of $G$. I am intressted in tilting ...

**4**

votes

**1**answer

209 views

### The existence of a finite dimensional Lie algebra with a given symmetric invariant metric

The question is motivated by a more broad perspective in another MO post and here, but here we would like to understand a specific case (our question potentially connects to / is motivated b Quantum ...

**4**

votes

**1**answer

192 views

### Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$

Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From ...

**2**

votes

**0**answers

198 views

### Commutators for quantum Lie algebras

Can the usual definition of a Lie algebra via commutators be simply adapted
to quantum Lie algebras? Graphically you have the IHX scheme, with the X
being a virtual crossing (so to say). Does it ...

**3**

votes

**2**answers

299 views

### Complete classification of six dimensional non-semi simple Lie algebra

I would aim to know the complete classification of 6 dimensional non-semi simple Lie algebra (here the dimension stands for the generators; or the dimension $\leq 6$).
In this paper, in page 7, it ...

**-1**

votes

**1**answer

126 views

### Highest weights of irreducible components of tensor product of irreducible sl(3)-module [closed]

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows:
For each weight $\mu$, let $L(\mu)$ be the irreducible ...

**5**

votes

**2**answers

247 views

### Decomposing representations of GL(n,F_q) induced from certain kinds of parabolics

The answer to the question below is almost certainly known to the representation theorists; in fact, I'm pretty sure it can be extracted from Green's paper "The characters of the finite general linear ...

**2**

votes

**1**answer

184 views

### Question about a proof in Graham and Lehrer's “Cellular algebras”

I'm sorry if this question is too basic for MO. I'm reading a paper by Graham and Lehrer "Cellular algebras" and have trouble understanding one step in a proof of a crucial theorem. I suppose that the ...

**3**

votes

**3**answers

382 views

### Definition of Hecke operators

I am confused about the definition of Hecke operators. It will be great if someone provides some references.
Shimura's 'Arithmetic Theory of Automorphic forms' says: Let $\Gamma$ be acting in the ...

**4**

votes

**3**answers

379 views

### Reg the motivation behind Lusztig-Vogan bijection

Let $G$ be an algebraic group. Choose a Borel subgroup $B$ and
a maximal Torus $T \subset B$. Let $\Lambda$ be the set of weights wrt $T$ and let $\mathfrak{g}$ be the lie algebra of $G$.
Now, ...

**1**

vote

**1**answer

95 views

### Fell topology in terms of distributions

Question: Can the Fell topology be expressed in terms of the distributions of the the tracial states of a unitary representations, that, is $\pi_j \rightarrow \pi$ if and only if $tr\; \pi_j ...

**3**

votes

**3**answers

230 views

### Finding a character of height zero

My character theory is rather weak, so excuse me if this is a triviality.
I have read on the encyclopedia of maths that for any group $G$, every block of $G$ contains an irreducible character of ...

**2**

votes

**1**answer

204 views

### Is Mazur's deformation ring R integral?

Consider the absolutely irreducible Galois representation
$\overline{\rho} \colon G_{\Bbb Q} \to {\mathrm{GL}}_2({\Bbb F}_p)$. We apply the Mazur's deformation theory on the lift ${\rho} \colon ...

**13**

votes

**2**answers

365 views

### Does base extension reflect the property of being isomorphic?

Let L/K be a (separable?) field extension, let A be a finite dimensional algebra over K, and let M and N be two A-modules. Let $A' = L \otimes_K A$ be the algebra given by extension of scalars, and ...

**0**

votes

**1**answer

218 views

### Iwasawa theory for Mazur's deformation ring R

The ideal class group $\mathrm{Cl}({\cal O}_K)$ and Mazur's deformation ring $R(\overline{\rho})$ for a number field $K$ are said to be similar to each other.
Let ${\Bbb Q}_{\infty}$ be the unique ...

**3**

votes

**1**answer

103 views

### Differential operators and commuting actions

We have a smooth space $X$ with an action from $G_1 \times G_2$ on it; we also have a differential operator $P \in \mathscr{D}(X)$. If $P$ takes $G_1 \times G_2$-invariant functions to $G_1 \times ...

**6**

votes

**0**answers

91 views

### Cominuscule property of nilpotent orbits

Let $G$ be a complex reductive Lie group, $G/P$ a flag manifold,
and $\Phi: T^* G/P \to {\mathfrak g}^*$ the moment map. So $\Phi(T^* G/P)$ is the closure of a nilpotent orbit.
Lots of classes of ...

**3**

votes

**3**answers

102 views

### For centralizer subgroups, is the endomorphism ring of a restriction generated by endomorphisms and the centralized element?

In some recent doodlings, I got myself to the point where what I was trying to understand would work out if the following claim were true:
Let $G$ be a group, $g\in G$, and $\rho:G \to ...

**5**

votes

**1**answer

171 views

### Decomposition into irreducibles of the representation $L^2(SL_2(\mathbb{C})/\Gamma)$ for $\Gamma$ geometrically finite

I am trying to understand the decomposition
$$L^2(SL_2(\mathbb{C})/\Gamma)=\oplus_i C_i \oplus V_{temp}$$
where $C_i$ are complementary series representations corresponding to eigenfunctions of the ...

**6**

votes

**1**answer

104 views

### Is the kernel of the Bohr compactification minimally almost periodic provided that it is cocompact?

Let $G$ be a locally compact (second countable) group and let
$$
G_0 = \cap \{ \ker\pi : \pi \text{ is a continuous finite-dimensional unitary representation of } G \}.
$$
This is the kernel of the ...

**1**

vote

**1**answer

101 views

### Specht polynomials for dihedral groups

The representation theory of the symmetric group is best understood via the Specht polynomials. In wonder how this works for other finite reflection groups, such as dihedral groups. Are the similarly ...

**1**

vote

**2**answers

194 views

### On size of Hecke algebras.

Let $G$ be a subgroup in $SL_2(\mathbb{Z})$ and $S_k(G)$ be the space of cusp (automorphic?) forms invariant by any element of $G$ of weight $k$.
Question 1: Generally for two arithmetic subgroups ...

**5**

votes

**2**answers

309 views

### Representation-theoretic operations on modular forms

Let $A$ and $B$ be Hecke eigenforms of some weight $k$ and level $N$. We know that there are irreducible representations $\rho_a$, $\rho_b$ of the absolute Galois group of $\mathbb{Q}$ whose trace of ...

**0**

votes

**2**answers

180 views

### 0-dimensional Gorenstein local ring.

Assume the following condition for the ring T = F_p[[X,S]]/I:
Condition 1. T is NOT a zero ring.
Condition 2. I is generated by 3 elements of F_p[[X,S]], but NOT by 2 elements.
Then, is T a ...

**3**

votes

**2**answers

146 views

### differences between character distributions of supercuspidal representations and others

Let $G$ be a $p$-adic linear reductive group. For an irreducible admissible smooth representation $\pi$ of $G$, there is a distribution $\Theta(\pi)$, called the character distribution, attached to ...

**4**

votes

**1**answer

229 views

### Is a group scheme determined by its category of representations?

More precisely, let $G$ be an affine group scheme over a field $k$, $Rep_k(G)$ be the category of finite dimensional representations of G, and $\omega_0$ be the forgetful functor from $Rep_k(G)$ to ...

**5**

votes

**0**answers

113 views

### Find all Linear spaces on an orbit (by a connected linear algebraic group)

Suppose $V$ is a vector space over $\mathbb{C}$ and $G\subset \textrm{GL}(V)$ is a connected linear algebraic group.
Consider the orbit closure $Y = \overline{G.\mathbb{P} L}$, for a subspace ...

**3**

votes

**1**answer

237 views

### Maximal Submodule of a Verma Module

Let $\mathfrak{h}$ be a Cartan subalgebra of a $\mathbb{C}$-semi simple Lie algebra $\mathfrak{g}$. Given $\lambda \in \mathfrak{h}^*$, $M(\lambda)$ the Verma module of highest weight $\lambda$ and ...

**3**

votes

**0**answers

125 views

### Generalization of Frobenius formula involving Macdonald polynomials

Given a vector $\vec k=(k_1,k_2,\cdots)$ with $k_i$ are non-negative integers, the Newton polynomial $p_{\vec k}(x)$ is defined as
\begin{equation}
p_{\vec k}(x)=\prod_{j=1}^n p_j^{k_j}(x)~,
...

**4**

votes

**1**answer

365 views

### Checking irreducibility

This is related to this question. Suppose I have an $n$-dimensional representation of a finitely generated group, and I want to know whether it is absolutely irreducible. This can, of course, be done ...

**5**

votes

**1**answer

168 views

### Stratifications and Filtrations of the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r=G(\mathcal{\mathbb{C}((t))})/G(\mathcal{\mathbb{C}[[t]]})$$ be the affine Grassmannian of $G$. We know that ...

**3**

votes

**1**answer

196 views

### Conductor of a representation of a $p$-adic group

Let $G$ be a connected split reductive group over $\mathbb{Z}$. Let $F$ be a local non-Archimedean field. Let $\rho$ be an irreducible smooth representation of $G(F)$. How does one define the ...

**9**

votes

**2**answers

351 views

### Cohomology ring of a flag variety and representation theory

I'm interested in the cohomology ring $H^*(G/B)$ of a flag variety $G/B$, where $G$ is a complex semi-simple Lie group and $B$ the Borel subgroup. Borel (1953) showed that this ring is isomorphic to ...

**0**

votes

**1**answer

153 views

### $I/N$ is finitely presented module

Let $R$ be a commutative ring and $N = Nil(R)$ the set of its nilpotent elements. Suppose that $N$ is a divided prime ideal, i.e. for any ideal $I$ of $R$ either $I \subseteq N$ or $N \subseteq I$.
...

**7**

votes

**2**answers

232 views

### Realizing a subgroup of a Lie group as a stabilizer subgroup

Let $G$ a compact semisimple Lie group, $H$ a subgroup of $G$. Is it always possible to find an irreducible representation $R$ of $G$ such that the stabilizer of an $x\in R$ is "locally isomorphic" to ...

**3**

votes

**1**answer

308 views

### Is there any groups $G$ with the property $(*_d)$?

Let $G$ be a finite group of even order has only one non-principal irreducible character $\chi$ of degree $d$, $d\in \mathbb{N}$, with the following property (we name it $(*_d)$):
$(*_d)$: There ...

**4**

votes

**1**answer

148 views

### Discontinuous representations of GL(n,C) in ZF

Discontinuous linear representations of $GL(n,\mathbb{C})$ can be obtained from the so-called "wild" (field) automorphisms of $\mathbb{C}$; but these wild automorphisms in turn require some choice to ...

**1**

vote

**0**answers

77 views

### faithful modules of algebraic group

Let $G$ be a linear algebraic group over a field $k$. $k[G]$ is the
coordinate ring of $G$. $k[G]^{*}$ is the dual algebra of the
coalgebra $k[G]$. $H=k[G]^{\circ}$ is the finite dual of the Hopf
...

**2**

votes

**0**answers

115 views

### Do you know any clear classification of groups in which there would exist a unique non-linear character of a given degree?

According to
Lev Kazarin, On Thompson’s Theorem, Journal of Algebra 220, 574–590 (1999)
we know that:
[Corollary 5.3]:Let $$cd(G)=\{\chi(1)|\chi\in Irr(G)\}=\{1,f_1,\dots,f_n,d\}, \;\;n\gt0,$$
...

**3**

votes

**1**answer

183 views

### On a unitary automorphic representation

I sometimes come across this notion called "unitary automorphic representation". But I have never seen the precise definition. When they say $(\pi, V)$ is a unitary automorphic representation of a ...

**6**

votes

**5**answers

463 views

### Is a unitary representation always semisimple?

I have been reading the online lecture notes by Fiona Murnaghan
http://www.math.toronto.edu/murnaghan/courses/mat1197/notes.pdf
The first lemma in p.35 says that every unitary representation of ...

**4**

votes

**1**answer

149 views

### When does the derived subgroup of $G(F)$ contains the $F$-points of unipotent subgroups of $G$

Let $F$ be a local field of characteristic $0$ and $G$ a connected split reductive group over $F$.
Let's look at the derived groups. We have $(G(F),G(F)) \subset (G,G)(F)$ and this inclusion is of ...

**5**

votes

**2**answers

264 views

### Strata of the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and denote by $\mathcal{G}$ its affine Grassmannian. Fix a maximal torus $T\subseteq G$. We know that $\mathcal{G}$ ...

**2**

votes

**0**answers

110 views

### irreducibility of exterior powers

Suppose I have a subgroup $H$ of $GL(V)$ such that $H$ acts irreducibly on all the exterior powers of $V$. Is there any sort of characterization of such things? (I am intentionally not specifying the ...

**4**

votes

**1**answer

341 views

### Homotopy of quivers

The matrix ring $k^{n\times n}$ can be realized in many ways as a quotient of a path algebra: For example choose the quiver $1\leftrightarrows 2 \leftrightarrows \cdots \leftrightarrows ...

**1**

vote

**0**answers

101 views

### How to decompose tensor products of simple modules for algebraic groups in GAP (or similar) [closed]

Is it possible to decompose tensor products for algebraic groups (in characteristic zero) in GAP?
I know that GAP has a Littlewood-Richardson rule function and is very good for character tables of ...

**4**

votes

**3**answers

366 views

### Clifford's Theorem with all its aspects in modern language, looking for a textbook

I am looking for a (more or less) introductory textbook on representation theory that contains the full contents of Clifford's paper "Representations Induced In An Invariant Subgroup" in modern ...

**5**

votes

**1**answer

220 views

### Artin representations in Serre's book 'local fields'

Let $K$ be a complete local field with discrete valuation, and let $L/K$ be a finite Galois extension. Use $G=Gal(L/K)$ to denote the Galois group.
In Serre's book 'local fields', chapter 6, a ...

**2**

votes

**0**answers

67 views

### Weyl group invariants of the representation ring of a split torus

Let $G$ be a semisimple split algebraic group, $T$ its split maximal torus and $W$ corresponding Weyl group. Let $T^*$ denote the character lattice of $T$ and $\Lambda$ denote the weight lattice, so ...

**5**

votes

**1**answer

135 views

### Pro-$l$ Sylow action in a primitive representation of inertia over $\overline{\mathbb{F}}_l$

Let $K$ be a nonarchimedean local field of residue characteristic $p \neq l$ and let $I_K$ be the inertia subgroup of its absolute Galois group. Let $V$ an irreducible representation of $I_K$ over ...