All Questions
304 questions with no upvoted or accepted answers
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95
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Injection of $G(k)/Z(k)$ into $(G/Z)(k)$
In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? ...
- 131
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64
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A variation of the dual group of the adjoint group
Let $\mathbf{G}$ be connected reductive group over a $p$-adic field $F$. Denote by $\mathbf{Z}$ the center of $\mathbf{G}$, and $\mathbf{A}$ the maximal split torus of $\mathbf{Z}$ (also called the ...
- 709
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50
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Do parabolic/Levi pairs admit dynamic descriptions over disconnected base?
In Gille, Thm. 7.3.1, it is proven that given a reductive group scheme $G \to S$ over a connected base $S$, every parabolic-Levi pair $(P, L)$ over $S$ admits a dynamic description, i.e. is of the ...
- 605
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102
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Does $\tilde{\mathrm E}_{6,3}^{(2)6}$ exist over a p-adic field?
Does a form of $\tilde{\mathrm E}_6^{(2)}$ with this Satake-Tits diagram exist over a p-adic field?
- 2,773
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107
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Structure theory of Schubert varieties (extend results from semisimple groups to reductive)
The lecture notes Borel–Weil–Bott theorem and geometry of Schubert varieties by Shrawan Kumar present a concise summary of major results on cohomology of flag varieties $G/B$ for $G$ semisimple, ...
- 6,006
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88
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Bad primes of twists of modular curves $X_E^{-1}(p)$
I am interested in a good reference for reading about primes of bad reduction for the modular curve $X_E^{-}(N)$, which is a twist of $X(N)$ parametrizing all elliptic curves $E'$ whose $N$-torsion ...
- 637
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150
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Relative compactification without resolutions of singularities
Let $Y$ be a smooth proper variety over a field $k$, let $X$ be a smooth variety over $k$, $U\hookrightarrow X$ the complement of a strict normal crossing divisor and $\phi\colon U\to Y$ a map. By ...
- 309
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88
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Relative position of flags for the general linear group
This question is related my question in An example of a Deligne–Lusztig variety for a general linear group, and the obtained answer.
Situation
I am working with the general linear group. Specifically, ...
- 153
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98
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Connected stabilisers for actions of reductive groups
Let $G$ be a connected split reductive group over a field $k$ acting on a variety $X$ over $k$. For each $x\in X$, let $G_x$ be the stabiliser. In general, $G_x$ may be disconnected.
Now suppose $G$ ...
- 2,751
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45
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Eigenvalues of orthogonal group element
Let $q$ be a quadratic form over a nonarchimedean local field $F$, and let $\operatorname{O}(q)$ be the corresponding orthogonal group. Let $g\in\operatorname{O}(q)$ be semisimple.
Can we know ...
- 833
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73
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Minimum level principal congruence subgroup coming from neat open compact subgroup
For a connected reductive group $G/\mathbb{Q}$ what is known about the minimum level such that the respective principal congruence subgroup is the intersection of a neat open compact subgroup of $G(\...
- 11
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177
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Restriction of a line bundle on $G/B$ to a fibre which is isomorphic to $\mathbb{P}^1$
Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$, Borel $B \supset T$ and Weyl group $W$. Set $X:=G/B$ and $C_w:=BwB/B \subset X$ for $w \in W$ the ...
- 473
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114
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Maximal compact subgroups of p-adic orthogonal groups
Let F be a non-archimedean local field, and G be split orthogonal groups of odd degrees $\geq$ 3.
In this setting, my question is;
Is there explicit descriptions of maximal compact subgroups of G?
I ...
user163485
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59
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Distributivity property for smooth parabolic induction
Let $G$ be a reductive group over a local field $k$ of characteristic zero with maximal split torus $T$, Weyl group $W$, Borel $ B$ and a parabolic subgroup $P$ such that $P\supset B \supset T$. ...
- 473
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92
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The kernel $K(x,y)$ as an integral over Eisenstein series for $\operatorname{GL}_2$
Let $G = \operatorname{GL}_2$, $f \in C_c^{\infty}(G(\mathbb A)/Z(\mathbb A))$, and $V = L^2( G(\mathbb Q)Z(\mathbb A)\backslash G(\mathbb A))$ (trivial central character). Then the operator $R(f)$ ...
- 6,180
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88
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$H(\mathbb A)^0/H(k)$ is homeomorphic to a closed set in $G(\mathbb A)/G(k)$
I'm reading Godement's paper Domaines fondamentaux des groupes arithmetiques and am confused about a part in a proof. The setting is $k$ is a number field, $\mathbb A$ is the ring of adeles of $k$, $...
- 6,180
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281
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Relation between reductive homogeneous spaces and reductive groups
To start, I would like to note that my background on Lie algebras is quite basic, so this question might be trivial when seen from a Lie algebra perspective, which I lack.
We have the concept of a ...
- 177
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51
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Do we have $\cap_{P\in S}\left(\mathcal{O}_P+(1-\tau)(K)\right)=\left(\cap_{P\in S}\mathcal{O}_P\right)+(1-\tau)(K)$?
Let $\mathbb{F}$ be a finite field and let $q$ be its number of elements. Let $(C,\mathcal{O}_C)$ be a geometrically irreducible smooth projective curve over $\mathbb{F}$ and let $K=\mathbb{F}(C)$ be ...
- 1,479
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91
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Why does norm map the $\sigma$-conjugacy classes to the conjugacy classes?
Let $E/F$ be a cyclic extension of order $\ell$ (not assumed prime) of fields of characteristic $0$, and $\Sigma$ its Galois group; we denote by $\sigma$ a generator of $\Sigma$. We denote by $G(E), G(...
- 479
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111
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Reference request: Commutator relations for the exceptional group F4
Is there any standard reference for the commutator relations for the exceptional group of type $F_4$?
If this question is not appropriate here, please let me know and I will delete it.
Thanks in ...
- 960
1
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246
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Frobenius twist of a field
Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
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82
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Connection between global and local notions of a cuspidal representation
Let $k$ be a number field, and $G$ a connected, reductive group over $k$. Let $\omega$ be a unitary character of $Z_G(\mathbb A_k)/Z_G(k)$. An irreducible subspace $(\pi, V)$ of $L^2(G(k) \backslash ...
- 6,180
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0
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63
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Question on the proof that the Jacquet module preserves admissibility
Let $P = MN$ be a parabolic subgroup of a reductive group $G$ over a $p$-adic field. For $(\pi,V)$ an admissible representation of $G$, the Jacquet module $(\pi_N,V_N)$ is defined by the action of $\...
- 6,180
1
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100
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Embedding of discrete series
Let $G$ be a simple Lie group with equal rank; namely, the rank of $G$ equals that of its maximal compact subgroup. Let $G'$ be a reductive subgroup of $G$ with equal rank. If $\tau$ is a discrete ...
- 951
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141
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Do we have $K \cap P = (K \cap M)(K \cap N)$?
Let $G$ be a connected, reductive group over a $p$-adic field $k$, let $P$ be a parabolic subgroup with Levi $M$ and radical $N$. Let $K$ be a maximal open compact subgroup of $G$ in good position ...
- 6,180
1
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86
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The convergence of the factor $\gamma(P)$ in the Iwasawa decomposition
Let $G$ be a connected, reductive group over a $p$-adic field $k$, $A_0$ a maximal split torus of $G$, and $P = MU$ a parabolic subgroup with $M$ containing $A_0$. Let $\overline{P} = M \overline{U}$ ...
- 6,180
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468
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Pairing half the sum of the roots with a simple coroot
I was calculating something with the root system $A_n$ and I think there might be a more general principle at work.
Here is the example: let $G = \operatorname{GL}_5$, with maximal torus $T$ and ...
- 6,180
1
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132
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Is the root cone is contained in the weight cone?
Originally posted on math.stackexchange. Let $G$ be a semisimple algebraic group over a field $k$. Let $A_0$ be a maximal split torus of $G$, and $\mathfrak a_0 = \operatorname{Hom}(X(A_0), \mathbb ...
- 6,180
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230
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Group schemes and Hyperspecial maximal compact subgroups
Let $F$ be a number field. For each non-archimedean place $v$ let $O_v$ denote the ring of integers. Let $G$ be a connected linear algebraic group defined over $F$. Consider the set of sequences $(K_v)...
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53
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A normalized embedding $\mathbb C \rightarrow \mathfrak a_M^{\ast}$ via $\tilde{\alpha}$
Let $G$ be a connected, reductive group over a field $k$. Let $S$ be a maximal $k$-split torus of $G$ with Weyl group $W$, $\Delta$ a set of simple roots of $S$ in $G$, and $P = MN$ a maximal ...
- 6,180
1
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348
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rigid analytic geometry positive characteristic
I am a beginning graduate student. I have the following basic question I am very confused about:
Suppose $C$ is a smooth geometrically irreducible curve over a finite field $\mathbb{F}_q$, $q=p^m$, $...
- 147
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956
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Iwasawa decomposition and compact subgroups
Let $G$ be the $k$-points of a connected, reductive group $\mathbf G$ over a local field $k$. I have heard several statements about compact subgroups and Iwasawa decomposition, mostly in the context ...
- 6,180
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187
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Unitary dual of the Heisenberg group over non-archimedean local fields of characteristic two
What is the unitary dual of the Heisenberg group over non-archimedean local fields k of characteristic two? This is well-known for the real Heisenberg group and in fact, when local fields have ...
- 1,702
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149
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Smoothability of stable curves in mixed characteristic
Let $R$ be a complete DVR with residue field $k$ algebraically closed of characteristic $p$ and fraction field $K$ of characteristic zero. Let $C$ be a stable curve (in the sense of Mumford-Deligne) ...
- 2,323
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109
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Reference Request: (Borel) Iwahori Spherical Representations
I was told that Borel had a result about Iwahori Spherical automorphic representations being upper-triangular (/semistable). Where can I find this?
- 1,629
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166
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Conjugacy scheme, fppf versus GIT
I would be glad to have some guidance in the following.
Let $k$ be an algebraically closed field. Let $G$ be a connected reductive group over $k$. Denote by $\mathfrak{c}$ the Zariski spectrum of the ...
- 5,562
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231
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Product of standard parabolic subgroups
Consider a quasi-split reductive group $G$ over a field $k$. Let $B$ be a Borel subgroup of $G$, and let $P, Q$ be two parabolic subgroups of $G$ that contain $B$. Is the product set $PQ = \{xy| x \in ...
- 11
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120
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Vanishing theorems that work in positive characteristic
Let $X$ be a smooth projective variety over a field of characteristic $p>0$ of dimension at least $2$. I am looking for some examples when $H^2(\mathcal{O}_X)$ vanishes. Is there any standard way ...
- 1,981
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Invariant vectors in supercuspidal representations of GL_2(Zp)
Let $o$ be the ring of integers in a local field $F$ with prime-ideal $p$. Let $K$ be either $GL_2(o)$ or the normalizer of the Iwahori subgroup. Let $\sigma$ be a representation of $K$ times the ...
- 11.2k
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How to Construct a ''Nice'' Birational Model in Characteristic $p>0$?
Let $X={\rm Spec} A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ is a prime Weil-divisor on $X$.
Now,...
- 165
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220
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Dimension of faithful irreducible representations of $\mathbb{Z}_q\rtimes \mathbb{Z}_{p^2}$ in characteristic p,q
Let $p,q$ be primes s.t. $q=np+1$. Denote $m=p^2$. Then $\mathbb{Z}_p$ acts non-trivially on $\mathbb{Z}_q$, so we have a non-abelian semi-direct product $\mathbb{Z}_q\rtimes \mathbb{Z}_m$, with the ...
- 407
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192
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"Higher" Tangent spaces in char-p geometry - definition?
Hi, everyone!
I have some construction that requires exact definition.
I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...
- 1,422
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238
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Classification of fibres in pencils of curves of genus two
For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
- 11
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157
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On closed abelian reductive subgroups of Real reductive groups
Hello everybody. I would first like to apologise for the basic question; I'm not expert on Lie Theory. Can someone please help me with the following questions
Let $\mathrm{G}=\mathrm{K} \exp(\...
- 11
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190
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About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 348
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64
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What does it mean for a linear algebraic group to act reductively
I was reading this paper by Baues and on page 918 he mention that $S$ acts reductively on the cochain complex and on page 919 again he mention the word "Since $T$ acts reductively on the complex.....
- 191
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0
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74
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Embeddings of unitary groups over $\mathbb{Q}$
$\DeclareMathOperator\GU{GU}$$\DeclareMathOperator\GL{GL}$I'm a bit confused by the following situation:
suppose we have an Hermitian vector space $V=K^3$ of matrix $$
J=\begin{pmatrix}& & \...
- 91
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0
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134
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Tempered representations and unramified principal series
For $V$ a tempered representation of connected reductive group over a local field of characteristic zero. I want to show that for an Iwahori subgroup $B$, the set of fixed points $V^B\neq 0$, thereby ...
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129
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Splitting of simply connected algebraic group
Let $k$ be a number field and let $G$ be a connected semisimple, simply connected algebraic group defined over $k$. Let $k'$ be a finite Galois extension over which $G$ splits. By the Chebotarev ...
- 223
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459
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Clarification on the definition of a smooth affine scheme over an integral domain
$\DeclareMathOperator{\Spec}{Spec}$
The following is from Bruhat and Tits article Groupes Reductifs sur un Corps Locale II. $A$ is an integral domain. Here $A$-scheme means "affine $A$-scheme," and $...
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