For questions about groups which have additional structure as a algebraic varieties (the vanishing sets of collections of polynomials) which is compatible with their group structure. Consider using with the (group-theory) tag.

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4
votes
1answer
95 views

How to prove that $\zeta*\zeta=\zeta$?

Let $F$ be a non-archimedean local field and $\mathcal{O}_F$ the ring of integers in $F$. Let $G_F=GL_2(F)$. Let $\pi_i$, $i=1,\ldots,n$,be non-equivalent finite dimensional irreducible ...
4
votes
1answer
23 views

Is $GL_2(\mathbb{Z}_p)$ a maximal compact subgroup of $GL_2(\mathbb{Q}_p)$?

Let $\mathbb{Q}_p$ be the set of all p-adic numbers and $\mathbb{Z}_p$ the set of all p-adic integers. Is $GL_2(\mathbb{Z}_p)$ a maximal compact subgroup of $GL_2(\mathbb{Q}_p)$? Thank you very ...
0
votes
0answers
3 views

Maximal tori in SL_2(k) with k an algebraically closed field of characteristic two

Suppose that $k$ is an algebraically closed field of characteristic two and that $T$ is a maximal torus of $\mathrm{SL}_{2}(k)$. Is the Lie algebra of $T$ necessarily equal to the centre of ...
2
votes
0answers
23 views

Simple connected semi-simple group

Here is a question form springer's book Linear Algebraic Groups, 8.4.6(6) Let $G$ be semi-simple and simple-connected, $P$ a parabolic subgroup of $G$ with Levi group $L$, Prove the commutator ...
3
votes
2answers
56 views

Proving the associativity of a monoid with $a \circ b = a+b-ab$

For university, it was my excercise to proof the associativity of the monoid $$H=(\mathbb{Q},\circ)\text{ with } a \circ b := a+b-ab\quad(a,b \in \mathbb{Q})$$ The excercise instructor gave us the ...
1
vote
1answer
33 views

For an element $x$ in an algebraic group $G$, why do we have $\mathscr{L}(C_G(x))\subset\mathfrak{c}_{\mathfrak{g}}(x)$?

I'm reading Humphreys' Linear Algebraic Groups, trying to understand the following argument found on the top of pg. 76. Let $G$ be an algebraic group over some field $k$, with $x\in G$. Let ...
1
vote
1answer
83 views

Elements of order 3 in $PGL(4,\mathbb{R})$

I need to classify all elements of order 3 up to conjugation in $PGL(4,\mathbb{R})$. It's sufficient to give a representative of each conjugacy class. My thoughts: consider instead $GL(4,\mathbb{C})$ ...
0
votes
1answer
31 views

Orthogonal invariants of an irredubile GL-representation

Let $n\in 2\mathbb Z$ be an even number. Let $G=\operatorname{GL}_n(\mathbb{C})$ and $V_\lambda$ the irreducible complex $G$-module corresponding to the partition ...
2
votes
4answers
47 views

Index of $\langle a^4\rangle$ in the group $\langle a\rangle$

Let $a$ be an element of order 30 in a group $G$. What is the index of $\langle a^4\rangle$ in the group $\langle a \rangle$? The answer is 2, however I have no idea on how to obtain that answer. ...
1
vote
2answers
36 views

Does the quotient of an algebraic group by its neutral component always split?

Let $G$ be a complex algebraic group. Let $G^\circ\subseteq G$ be the connected component of $G$ which contains the neutral element $1\in G$. Then, it can be shown that $G^\circ\trianglelefteq G$ is a ...
1
vote
1answer
27 views

A question about reductive groups's structure.

There is a flaw somewhere in the following argument but I can't track it. Take a reductive connected affine algebraic group $G$ : by definition, its unipotent radical $R_u(G)$ is trivial. One have ...
2
votes
2answers
69 views

Dimension of a $G$-variety $X$ that is a finite union of $G$-orbits

Suppose that $G$ is an algebraic group acting on a variety $X$, and $X$ is a finite disjoint union of $G$-orbits $\mathcal{O}_i$, $i=1,\ldots,n$, under this action. Is it true that the dimension of ...
2
votes
1answer
56 views

Confusion about the quotient $G/B$

Let $G$ be an affine, complex, reductive algebraic group and $B$ a Borel of $G$. I have seen and understood the proof that $G/B$ is projective. Now, on the other hand, I have made the following ...
4
votes
1answer
63 views

Linearization of a group action: why the map is equivariant?

I'm using Dolgachev's book on invariant theory to learn linearizations of group actions. Here is a sketch of main construction: let linear algebraic group $G$ act on a quasi-projective variety $X$, ...
1
vote
1answer
32 views

Torus orbit closures and rank-1 subtori

Suppose I have a connected complex torus $K$ acting on a quasi-affine complex variety $X$. Suppose also that I have $p,q\in X$ such that the orbit $Kq$ is closed in $X$ and $q\in ...
2
votes
0answers
51 views

finite normal subgroup

$G$ is a subgroup of finite index in $SL(n,Z)$, $n\ge 3$, $N$ is finite normal subgroup of $G$, then I want to know why $N$ is a normal subgroup of $SL(n,Z)$. More generally, $A$ is an arithmetic ...
2
votes
0answers
25 views

Semisimple part of a nilpotent connected affine algebraic group

These notes on affine algebraic groups mention the following theorem. Let $G$ be a connected nilpotent affine algebraic group (over an algebraically closed field $k$), and denote $G_s$ and $G_u$ ...
4
votes
0answers
79 views

Making the definition of dual root unambiguous

In 5.4 of his book Lectures on Invariant Theory, Igor Dolgachev introduces the dual of a root by requiring that $\check\alpha(t) f_\alpha(x) \check\alpha^{-1}(t)= f_\alpha(x)$ ...
2
votes
0answers
38 views

Confused about Borel-Weil theorem

I am trying to understand the Borel-Weil theorem, but I am very confused because of the different conventions used in different sources. I am especially confused about two things: (1) the definition ...
5
votes
1answer
97 views

What is the coordinate ring of $G/U$?

Let $G$ be an algebraic group and $U$ its subgroup consisting all upper triangular matrices. For example, $G=GL_n(k)$ and $U$ the subgroup consisting of all upper triangular unipotent matrices in ...
3
votes
1answer
25 views

Does the exponential map respect module actions?

Setup: Let $k$ be a field and $G \subseteq \mathrm{GL}_n(k)$ an algebraic group, reductive if that makes a difference. Let $\mathfrak g \subseteq \mathfrak{gl}_n(k)$ be the Lie algebra of $G$ with ...
1
vote
1answer
44 views

What is the group of $k$-rational points of an algebraic group?

Let $k$ be a field and $G$ a linear algebraic group over $k$. What is the group of $k$-rational points of $G$? By definition, $G$ is an algebraic variety. Suppose that $G$ is defined by polynomials ...
2
votes
1answer
70 views

The $\mathbb C((z))$-rational points of a complex semi-simple group $G$

By definition, if $R$ is a $\mathbb C$-algebra and $G$ is a $\mathbb C$-scheme then the set of $R$-valued points on $G$ is $G(R)=\hom_{\text{Sch}_{\mathbb C}}(\operatorname{Spec} R, G)$ In Ginzburg's ...
0
votes
0answers
50 views

Why Bruhat decomposition in $GL_n$ case is the Gauss decomposition?

Gauss decomposition of a matrix is also called LU decomposition. Let $A$ be a matrix. Then $A=LU$ for some lower triangular matrix $L$ and upper triangular matrix $U$. This can be obtained using Gauss ...
9
votes
0answers
122 views

a closed subset of an algebraic group with a constant tangent space is a coset

Let $G$ be an algebraic (not necessarily linear) group and let $Z \subset G$ be a Zariski closed irreducible subset. Since tangent bundle of $G$ is trivial, we may identify tangent spaces at all ...
1
vote
0answers
37 views

Questions about affine Weyl group and extended affine Weyl group for SL2.

Let $G=SL_2$. Then the Weyl group is generated by $s_1$. On page 3 of the lecture notes, it is said that the affine Weyl group is generated by $s_0, s_1$. (1) The element $s_0s_1$ can be identified ...
1
vote
0answers
68 views

P/B is isomorphic to the projective line $\mathbb{P}^1$

Suppose that $P \subset G$ is a parabolic subgroup containing a Borel subgroup $B$. Moreover, let $P$ be a minimal parabolic subgroup properly containing B, i.e., one corresponding to a single root ...
0
votes
0answers
29 views

What is a quotient of a building by a lattice?

For an algebraic group $G$, we may defined a building associated to $G$. Let $B = B(G)$ be the corresponding building. I don't understand much about the concept quotient $B/\Gamma$ of a building $B$ ...
4
votes
2answers
77 views

How many elements of order $k$ are in $S_n$?

I need to find how many elements of order $k$ are in $S_n$ (where $k \leq n$). So if $k$ is prime, it's easy: $k$ can't be the $\mathrm{lcm}$ of any integers besides itself and one's (which we're ...
2
votes
1answer
54 views

Factor/Quotient Group $M/\mathbb{Z}$

I've posted another question here a few days ago asking what a factor/quotient group is because I couldn't wrap my mind around it. Although I have an idea what it means, I still don't fully understand ...
3
votes
0answers
30 views

Closed subgroups of algebraic group have DCC?

Are there any infinite descending chains of closed subgroups of the general linear group over a field? More specifically, is my argument ok? Can you fill in some of the details? Prop: No. Proof: If ...
2
votes
1answer
33 views

Minimal parabolic subgroups of a reductive group - Bruhat type decomposition

Let $G$ be a reductive group, $B$ a Borel subgroup, $P$ a minimal parabolic subgroup having a Levi decomposition $P = UL$, let $\alpha$ be one of the two roots of $L$ relative to $T$, and $U_\alpha, ...
3
votes
1answer
62 views

Non-uniqueness of group structure for affine algebraic groups

We know that every abelian variety has a unique group structure, but in the affine case, is that every affine algebraic group has more than one (up to isomorphism) group structure?
1
vote
0answers
22 views

Baily Borel Compactification: choice of boundary

In Borel/Ji " compactifications of symmetric and locally symmetric spaces " the Baily Borel compactification of a locally symmetric space is defined as $$\Pi\backslash(X\coprod_{\bf{P}}X_{P,h})$$ ...
0
votes
0answers
28 views

References request about exponentials in Lie algebras.

I saw two formulas about Lie algebras. Let $G$ be an algebraic group over $k$ and $\mathfrak{g}$ its Lie algebra. For any $x \in \mathfrak{g}$, $a \in k$ and $g \in G$, we have $$ g \exp(ax) g^{-1} = ...
0
votes
0answers
34 views

Lie algebra of the unipotent radical of a standard parabolic subgroup in $GL_n$

Let $k$ be a field, and consider the algebraic group $G=GL_n(k)$. For any partition $n_1+n_2+\ldots+n_m=n$, we have a parabolic subgroup of the form ...
1
vote
0answers
29 views

Orbits and rational points in a $G$-variety

Let $K/k$ be a field extension, let $V_0$ be a variety over $k$, and let $V=V_0\times_k\mathrm{Spec}\;K$, so that we can speak of the $k$-rational points of $V$ as morphisms $\mathrm{Spec }\;k\to ...
0
votes
0answers
23 views

How does Weyl group acts on coroots?

We know that $W=N/T$ ($N = \{n \in G \mid nTn^{-1} = T \}$) acts on $T$ by $w(t)=wtw^{-1}$ (since $T$ is commutative, this action is well-defined). In the case of $GL_n$, by direct computation we know ...
4
votes
0answers
64 views

What is $\rho^{\vee}(-1)$?

What is $\rho^{\vee}(-1)$? By definition, $\rho^{\vee}$ is the sum of all positive coroots. I have some difficulty in computing $\rho^{\vee}(-1)$. For example, in the case of $SL_3$, all positive ...
1
vote
0answers
125 views

Possible small mistakes in Springer's *Linear Algebraic Groups*

In the 2nd edition of Springer's Linear Algebraic Groups, the proof of 16.2.2(i) (p. 271) begins with the assertion that the reductive group G is quasi-split over F iff the restriction map of ...
1
vote
0answers
43 views

Coxeter numbers for semisimple and reductive algebraic groups

I'd like to know how to define the coxeter number for semisimple and reductive algebraic groups. I know that for a simple algebraic group $G$, we can fix a maximal torus $T\subset G$, which acts on ...
6
votes
1answer
117 views

Geometric intuition for linearizing algebraic group action

I am reading an overview of geometric invariant theory and find myself stuck when we begin linearizing the action of an algebraic group on a variety. The definition given in my notes is that given an ...
0
votes
1answer
27 views

Rigidity of Diagonalizable Algebraic Groups

This question is about a result in the section 16.3 of the book Linear Algebraic Groups from Humphreys. The follow can be deduced from a proposition in the section 16.3 of the book: Corollary: Let ...
3
votes
0answers
32 views

Correspondence between unipotent and nilpotent elements

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. Let $\mathcal{U}(G)$ be the closed subvariety of unipotent elements of $G$, i.e., all elements whose ...
0
votes
0answers
22 views

Tate module of linear algebraic group

Let $G$ be a (smooth, connected, geometrically integral) commutative linear algebraic group over $\mathbf F_q$. Just as for abelian varieties, we can define the $\ell$-adic Tate module $$ T_\ell G ...
0
votes
0answers
46 views

Relations between $GL_n$, $SL_n$, $PGL_n$.

I have some questions about relations between semisimple and reductive groups. We know that $GL_n$ is reductive and $SL_n$ is semisimple. It is said that reductive groups are extensions of semisimple ...
2
votes
1answer
65 views

How to compute radical and unipotent radical of an algebraic group?

By definition, an algebraic group $G$ is reductive if its unipotent radical is $\{e\}$. The radical of an algebraic group is the identity component of its maximal normal solvable subgroup. The ...
1
vote
3answers
57 views

Examples of root, parabolic, and borel subgroups corresponding to roots

I'm interested in seeing a few examples of root, parabolic, and Borel subgroups given a specific reductive group $G$. Here is what I know. Let $G$ be a reductive algebraic group over an ...
2
votes
0answers
16 views

Example of algebraic group of type $G_2$

Can anyone point me to a concrete realization of a reductive algebraic group of type $G_2$ over a field of positive characteristic? I have some questions about how the adjoint action permutes certain ...
1
vote
0answers
34 views

why is Borel subgroup not nilpotent?

Let $G$ be a simple linear group group over an algebraically closed field $k$, and let $B$ be a maximal solvable subgroup. If things are happening over $\mathbb{C}$ then I know how to show that $B$ ...