# Questions tagged [rt.representation-theory]

Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

**3**

votes

**0**answers

65 views

### The arithmetic meaning of opers (if any)

Let $G$ be a complex, connected semi-simple Lie group, $G'$ its Langlands dual
group, $\mathrm{Bun}_G$ the moduli stack of $G$-bundles on a smooth projective curve $\Sigma$ over complex numbers, $\...

**1**

vote

**0**answers

72 views

### When is the Jacquet module semisimple?

Let $G$ be a reductive algebraic group over a $p$-adic field, $N$ be the unipotent radical of a parabolic subgroup $P$ of $G$, $M$ be a Levi subgroup of $P$ so $P = M \rtimes N$, and $\chi$ be a ...

**3**

votes

**0**answers

28 views

### Cartan determinants of minimal Auslander-Gorenstein algebras

Iyama and Solberg introduced minimal Auslander-Gorenstein algebras as algebras having finite dominant dimension ($\geq 2$) equal to the Goreinstein dimension in https://www.sciencedirect.com/science/...

**6**

votes

**1**answer

99 views

### Relative Dickson (trace) criterion for Jacobson radical?

In the following, all algebras are associative and unital. Let $J\left(A\right)$ denote the Jacobson radical of an arbitrary algebra $A$. Recall that this is defined as the set of all $a \in A$ such ...

**3**

votes

**1**answer

63 views

### Distance between Verma modules in certain “strongly” standard filtrations

On p. 128 of the book: Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$.
I quote: "......Delorme arrives at
vanishing criteria for Ext$^n(\mathcal{O})$ which are more ...

**2**

votes

**0**answers

24 views

### Extenstion functor on quotient map

Let $\rho$ be the half sum of positive roots in $\Phi^+$, $M_x$ be the Verma module with highest weight $x\cdot(-2\rho)$ and $L_w$ be the simple highest weight module with highest weight $w\cdot(-2\...

**4**

votes

**0**answers

54 views

### Stable equivalence and stable Auslander algebras

Let $A$ be a representation-finite finite dimensional quiver algebra and $M$ the basic direct sum of all indecomposable $A$-modules.
Recall that the Auslander algebra of $A$ is $End_A(M)$ and the ...

**4**

votes

**0**answers

54 views

### Derived equivalence between two exotic algebras

Let $A$ and $B$ be two connected finite dimensional quiver algebras having the same underlying quiver.
Question 1:
In case $A$ and $B$ have exactly one indecomposable projective non-injective $A$-...

**3**

votes

**0**answers

158 views

### Kazhdan–Lusztig polynomials in terms of Ext groups

Let $P_{x,w}$ be the Kazhdan–Lusztig polynomial, $\rho$ be the half sum of positive roots in $\Phi^+$, $M_x$ be the Verma module with highest weight $x\cdot(-2\rho)$ and $L_w$ be the simple highest ...

**4**

votes

**1**answer

109 views

### Typical and atypical modules for Lie superalgebras

There are two types highest weight representations for a Basic classical simple Lie superalgebra $\mathfrak{g}$ which are defined as typical (representation for which highest weight vector is the only ...

**2**

votes

**2**answers

72 views

### Symmetry of Casimirs of Lie algebras

The dimensions of the invariant tensors (Casimirs) of the simple Lie algebras are known, but I nowhere could find whether they are completely symmetric or antisymmetric with respect to an variable ...

**1**

vote

**0**answers

83 views

### Hilbert modular form as a representation of Hecke algebra

I am reading some notes by Snowden and I don't understand a sentence.
Clearly, if we have an appropriate $R = T$
theorem then we get a modularity lifting theorem, as $\rho$ defines a homomorphism ...

**4**

votes

**1**answer

166 views

### Do we have $G(\mathbb A_S) G(k) = G(\mathbb A)$ for sufficiently large $S$?

Let $G$ be a linear algebraic group over a number field $k$. If necessary, assume $G$ is connected and reductive. Let $\mathbb A$ be the ring of adeles of $k$, and $\mathbb A_S = \prod\limits_{v \in ...

**2**

votes

**1**answer

56 views

### Translation functor on parabolic Verma module

I want to prove that following proposition by using Theorems/propositions in Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$.
Define
$
\Lambda := \{\nu \in \mathfrak{h}^* ...

**4**

votes

**0**answers

43 views

### Intersection of components in Springer fibre of type A

From the standard results on Springer fibers of type A, we know that given a Springer fiber, say $\mathcal{B}_\lambda,$ its irreducible components are all equidimensional and parametrized by standard ...