# Tagged Questions

**6**

votes

**1**answer

117 views

### About G-modules with good filtrations

Let $k$ be an algebraically closed field of positive characteristic, and let $G$ be a reductive algebraic group over $k$ (for instance a classical group).
Let $V$ be a (rational) $G$-module. We say ...

**1**

vote

**0**answers

205 views

### Relationship between algebraic groups and Lie groups? [closed]

In the literature, e.g. in representation theory, there seems to be a passage from Lie groups to (linear) algebraic groups. It is clear, particularly over $\mathbb R$ and $\mathbb C$ that they are ...

**1**

vote

**0**answers

125 views

### Reductive Group Actions and Completion

Suppose that $A$ is a Noetherian (not necessarily commutative) $\mathbb{C}[h]$-algebra equipped with a rational action of an affine reductive group $G$, i.e., $A$ is a $\mathbb{C}[G]$-comodule and ...

**8**

votes

**3**answers

540 views

### Tensor products of two irreducible representations of reductive groups and their inclusions

Let $G$ be a reductive group and $\lambda$, $\mu$ and $\nu$ be dominant weights of $G$. Denote by $V_\lambda$ the irreducible representation of $G$ of highest weight $\lambda$. It seems to be true ...

**1**

vote

**1**answer

308 views

### What is the current status of representations of $GL_n(F)$ (and other algebraic groups)?

What is the current status of representations of $GL_n(F)$ (and other algebraic groups)?
When $F$ is a local field, the representations of $GL_n(F)$ are classified by Bernstein and Zelevinsky in ...

**7**

votes

**0**answers

248 views

### Connection between two theorems on character values?

In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...

**7**

votes

**1**answer

147 views

### Complexity of rational $\mathrm{GL}_{n(r)}$-modules

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G=\mathrm{GL}_n(k)$ for some natural number $n$. For any integer $r\ge 1$, let $G_{(r)}$ denote the $r$th Frobenius ...

**4**

votes

**1**answer

96 views

### Weyl group action on complexified Iwasawa decomposition

Let $G$ be a complex, reductive, algebraic group and let $G=KB$ be the complexified Iwasawa decomposition of $G$, see also [SW02]. Let $T$ be a maximal torus of $B$, therefore a maximal torus of $G$. ...

**4**

votes

**2**answers

224 views

### Gelfand pair and double coset decomposition

Let $F$ be a non-Archimedean local field with ring of integers $O$, $\pi$ be a uniformizer. Let $\tilde{G}$ be a connected algebraic group over $F$ and splits over $F$, fix a split maximal torus ...

**1**

vote

**0**answers

45 views

### Reference Help: Matsuki duality Orbits

I'm studying the Matsuki duality of $G_0$-orbits and $K$-orbits over a flag manifold $G/P$ where $G$ is semisimple complex Lie group and $P$ is a parabolic subgroup. I would like to study some ...

**6**

votes

**2**answers

272 views

### What can representations of affine Weyl groups do?

In Carter's Finite Groups of Lie type and Lusztig's Characters of Reductive Groups over a Finite Field, the representations of Weyl groups are helpful in finding the representations of algebraic ...

**0**

votes

**0**answers

235 views

### generators for derived category

Let $G$ be a algabraic group $G$ over a field $k$. We denote by $D^b(\mathrm{Repr}(G))$ the derived category of finite dimensional representations. Under what kind of assmumptions one has a generating ...

**2**

votes

**1**answer

118 views

### Is there any way to determine the “effeciency” of Jantzen's sum formula?

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a reductive algebraic group over $k$.
In order to determine the structure of the Weyl modules $V(\lambda)$, ...

**2**

votes

**1**answer

135 views

### Indefinite orthogonal groups over p-adics

Let $q$ be a rational quadratic form. How can we think of a Cartan decomposition of $O_q(Q_p)$? Is there a notion of Cartan involution for p-adic field, so that we can execute same process as we do ...

**3**

votes

**1**answer

190 views

### What is “special” maximal compact subgroup of algebraig group over local field?

Learning the theory of Langlands correspondence, I met the notion of "special" maximal compact subgroup of a (reductive) algebraic group over a local field.
Here, I think the word "compact" is used ...

**8**

votes

**0**answers

154 views

### Earliest use of the term “linearly reductive”?

Recently a number of MO questions have referred to a "linearly reductive group", usually in a way that is out of focus. It's unclear to me why this terminology is so popular, since over a field of ...

**1**

vote

**0**answers

127 views

### Algebraic characters and quasi-characters of reductive algebraic group over non-archimedean local field

Let $G$ be a reductive algebraic group over $F$, where $F$ is a non-archimedean local field.
Then $G(F)$ is a p-adic group.
Let $\Psi(G)$ be the lattice of algebraic characters.
Let $\Lambda_G$ be the ...

**6**

votes

**2**answers

240 views

### Whitney stratification and affine grassmanian

Let $G$ a simply connected group over $\mathbb{C}$ and $Gr:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ the affine grassmannian. By Cartan decomposition we have a partition of stratas indexed by ...

**1**

vote

**0**answers

117 views

### Restriction of the Steinberg representation

Let $G$ the group $GL(n,F)$ where F is a locally compact non Archimedean field, and $G^{0}$ the subgroup of $G$ consists of elements $g$ in $G$ such that $\det(g)$ in $\mathcal{O}_{F}^{\times}$, where ...

**2**

votes

**0**answers

69 views

### radical unipotent of a parahoric

Let $G$ a split connected reductive group over $\mathbb{C}$. $F=\mathbb{C}((t))$ and $\mathcal{O}$ the ring of integers.
Let $B$ a Borel subgroup and $I$ the corresponding Iwahori.
Let ...

**0**

votes

**0**answers

85 views

### intersection of Borel in wonderful compactification

Let $\overline{G}$ the wonderful compactification of an adjoint $G$ over an algebraically closed field $k$.
Let $B$ a Borel of $G$ and $w\in W$ an element of the Weyl group and $\overline{B}$ the ...

**1**

vote

**0**answers

80 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
...

**4**

votes

**1**answer

155 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 ...

**1**

vote

**0**answers

111 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 ...

**3**

votes

**1**answer

261 views

### A question on algebraic loop groops

Setup:
Let $\mathcal{K}=\mathbb{C}((t))$, $\mathcal{O}:= \mathbb{C}[[t]]$ and $G$ be a reductive algebraic group (over $\mathbb{C}$). Let further $\mathcal{K}_n$ denote the $\mathcal{O}$-ideal in ...

**7**

votes

**0**answers

225 views

### Higher-dimensional generalization of Pink's theorem

Pink's theorem in the title of the question refers to the main theorem of Pink's paper "Compact Subgroups of Linear Algebraic Groups" that appeared in Journal of Algebra (206) in 1998. It essentially ...

**6**

votes

**1**answer

151 views

### regular semisimple elements on spherical varieties

Let $(G,H_1)$ and $(G,H_2)$ be spherical pairs (i.e. $G$ is a reductive group, $H_i$ are its closed subgroups and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H_i$).
What can ...

**3**

votes

**0**answers

41 views

### points with small U stabilizer on a spherical variety

Let $(G,H)$ be a spherical pair (i.e. $G$ is a reductive group, $H$ is a closed subgroup and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H$). Let $U$ be the unipotent radical of ...

**4**

votes

**1**answer

266 views

### When does a group action on a k-algebra induce an algebraic action on the spectrum?

This question arose from my last question, which I considered answered - from the comments, however, it is obvious that the answer is only complete in characteristic zero, and I am trying to ...

**2**

votes

**0**answers

82 views

### integral stable conjugacy classes

Let $G$ be a semisimple simply connected group over $k$ algebraically closed field .
Let $\gamma,\gamma'\in G(k[[\pi]])$ that are generically regular semisimple on $G(k((\pi)))$.
We assume that ...

**1**

vote

**0**answers

135 views

### on the open bruhat cell

Let $G$ a connected reductive group and $S=U^{-}TU$ the open cell.
Do we have $G=\bigcup\limits_{g\in G}gSg^{-1}$?
And also if I assume that $G$ is adjoint and $\overline{G}$ is the de ...

**3**

votes

**3**answers

288 views

### Topological properties of $K$ orbits in $G/B$

I'll be working over the complex numbers.
Let $G$ be a connected reductive group, $\theta\colon G\to G$ an involution. Let $K=G^{\theta}$ be the fixed point subgroup. I am trying to track down ...

**4**

votes

**1**answer

197 views

### The Representation of $\mathrm{Sp}_{2n}$ of Dimension $2^n$ in characteristic 2

Let $G:=\mathrm{Sp}_{2n}$ be the simple algebraic group of simply connected type with root-system $C_n$.
Is there a way, to explicitly construct the
highest weight representation ...

**8**

votes

**1**answer

337 views

### Action of the endomorphism monoid on an irreducible GL-module

Let $G=\mathrm{Gl}_n(\mathbb C)$ and $V$ an irreducible $G$-module on which $G$ acts polynomially. In other words, the algebraic group action of $G$ on the affine space $V$ extends to an algebraic ...

**34**

votes

**3**answers

2k views

### What to do now that Lusztig's and James' conjectures have been shown to be false?

Lusztig and James provided conjectures for dimensions of simple modules (or decomposition numbers) for algebraic groups and symmetric groups in characteristic $p$. These conjectures have been ...

**2**

votes

**0**answers

240 views

### Representations of the orthogonal group O(n) vs representations of the special orthogonal group SO(n), over an arbitrary field

Let $O(n)$ and $SO(n)$ denote the split orthogonal linear algebraic group and its special subgroup, over some fixed field of characteristic not two.
I am looking for a reference that explains how to ...

**3**

votes

**2**answers

245 views

### Simple representations of products of algebraic groups

I am looking for a reference for the following assertion that I believe to be true. All representations are assumed to be finite-dimensional.
Let $G_1$ and $G_2$ be affine algebraic group schemes ...

**3**

votes

**0**answers

110 views

### Product of Fixed points and kernel of Frobenius morphism

If $G$ is a reductive algebraic group over an algebraically closed field of positive characteristic $p$, and $G$ is defined over the prime field, we have the Frobenius morphism $F: G\to G$, which for ...

**4**

votes

**1**answer

161 views

### Decomposing tensor products of modules for the orthogonal/symplectic groups in characteristic zero

I would like to know if there is a perfect analogue of the classical Littlewood-Richardson rule for decomposing tensor products of simple modules for the orthogonal/symplectic groups in characteristic ...

**4**

votes

**1**answer

261 views

### How to translate the representation theory of semisimple to reductive groups?

I am aware of the following question: Definitions of Reductive and Semisimple Groups
So let me phrase a precise question:
Is there a standard technique by which one can translate the ...

**2**

votes

**2**answers

374 views

### Equivariant Cohomology of a Complex Projective Variety

Suppose that I have a complex projective variety $X$ endowed with an algebraic action of a complex torus $T$. Suppose also that the set $X^T$ of fixed points is finite. I would like to relate the ...

**2**

votes

**1**answer

169 views

### On the $F$-rational points of the derived group of a connected reductive algebraic group

Let $F$ be a local non-archimedean field and let $G$ be a connected reductive algebraic group defined over $F$. Let $G_{der}$ denote the algebraic derived group of $G$; this is connected and ...

**1**

vote

**0**answers

48 views

### open immersion, affine grassmanian and negative loop group

Let $G$ a semisimple group over $k=\bar{k}$.
Let the $k$-indgroup, $LG^{-}\subset G(k[t^{-1}])$ be the kernel of the reduction. We know by Faltings that the multiplication map:
$LG^{-}\times ...

**2**

votes

**0**answers

106 views

### affine schubert cells and bruhat order

Let $G$ asimply connected group over $k=\bar{k}$, $B$ a Borel subgroup and $I$ the corresponding Iwahori in G(k[[t]]), $T$ a maximal torus and $K=G(k[[t]])$.
Let $\lambda\in X_{*}(T)^{+}$ a dominant ...

**2**

votes

**1**answer

128 views

### affine weyl group and affine schubert cells

Let $G$ a connected reductive split group over $k=\bar{k}$, $(B,T)$ a split Borel pair. Let $F:=k((t)))$. Let $\tilde{W}$ the extended Weyl group, $\tilde{W}=N_{G}(T(F))/T(O)$.
By Iwasawa ...

**2**

votes

**0**answers

178 views

### Generators of $Rep(G)$

Let $G$ be a reductive group over $\mathbb{C}$ and $Rep(G)$ the category of rational representations.
Is there a "nice" (let's say combinatorical) description of the generators of $Rep(G)$ as a tensor ...

**1**

vote

**1**answer

244 views

### weights and exceptional root systems

Let $G$ a simple simply connected group over $\mathbb{C}$ and $W$ his Weyl group.
Let $\lambda$ a minuscule or quasiminuscule weight.
For which types and for which weights do we have that:
$\forall ...

**3**

votes

**0**answers

125 views

### ideal generated by highest weight vectors

Let $S$ be a polynomial ring which carries the action of a semi-simple linear algebraic group $G$ (I'm interested in a product of $GL$'s). Take $S$ and $G$ to be over an algebraically closed field.
...

**0**

votes

**0**answers

85 views

### solve the singularities of parabolic orbits of schubert cells

Let G a semsisimple connect'ed group over $k$, $B$ a Borel and $P$ a parabolic subgroup of $G$ with Weyl group W_{P}.
For $w\in W_{P}\backslash W/W_{P}$, how can we solve the singularities of ...

**2**

votes

**1**answer

179 views

### On the Cartan decomposition of unitary group

Hello. I have some question on Cartan decomposition of unitary group, especially $U(2)$.
I am interested in local situation, that is p-adic or archimedian.
Let $F$ be a local field and $E$ be its ...