User ronggang - MathOverflow

arithmetic group over function fields and its fundamental domain

2013-04-23T18:21:29Z

Let $G$ be a semi-simple algebraic group defined over a global function field $K$. Let $S$ be a finite set of places of $K$. For a place $v$ of $K$ let $K_v$ be the completion under $v$. We take $K_S=\prod_{v\in S}K_v$ and $O={x\in K: x \mbox{ is integral in }K_v \mbox{ if } v\not \in S }$. It is well known that $G(O)$ is a lattice in $G(K_S)$. Are there any fundamental domain of $G(O)$ expressed in terms of Ziegel sets?

More precisely we want the following type result. If $H$ is semi-simple $\mathbb Q$-group, then it is proved in Theorem 15.5 of Borel's book "Introduction aux groupes arithmétic" that there exists a Ziegel set $F$ and a finite subset $C\subset G(Q)$ such that $FCG(\mathbb Z)=G(\mathbb R)$. To my understanding this result is not contain in the paper of Borel and Harish-Chandra 1962 on arithmetic groups.

It will be great if one can suggest some references (in English) about it.

local field and number field

2013-03-28T03:07:46Z

Let $K$ be a local field (locally compact topological field) of characteristic zero. Is it true that $K$ is isomorphic to the completion of a number field under some valuations? If yes, then how to prove it?

I ask this question since in a paper it is said that $k_v^*/(k_v^*)^2$ where $k_v$ is the completion of a number field at some place $v$ has order 1, 2, 4 or 8. From the structure theory of local fields this order argument is incorrect for general local fields of characteristic zero.

If not every local field of characteristic zero comes from the completion of some number field and the above claim about the order of $k_v^*/(k_v^*)^2$ is correct. Can some one prove it or give a reference for it?

a question on TITS' note "Reductive groups over local fields"

2012-09-03T16:50:05Z

This note appears in "Proceedings of Symposia in pure mathematics" vol.33 1979 part 1 pp. 26-69. The question will be about materials on page 31-32.

Let $G$ be a reductive algebraic group (not necessarily connected) defined over a local field $K$. We fix a maximal $K$-split torus $S$ of $G$ and take N(resp. Z) to be normalizer (resp. the centralizer) of $S$ in $G$. Let $X_*=Hom_K (Mult, S)$ (Resp. $X^*=Hom_K (S, Mult)$) be the group of cocharacters (resp. characters) of $S$. Let $V=X_*\otimes_{\mathbb Z}\mathbb R$. We fix a discrete valuation $\omega: K\to (-\infty, \infty]$. Let $\nu: Z(K)\to V$ be the homomorphism defined by $$ \chi (\nu (z))=-\omega (\chi (z)) \quad \mbox{for}\ z\in Z(K) \mbox{ and } \chi \in X^*(\mathbb Z). $$ Let $Z_c$ be the kernel of $\nu$. Then we have a short exact sequence of gorups $$ 0\to Z(K)/Z_c\to N(K)/Z_c\to N(K)/Z(K)\to 0 $$ where $N(K)/Z(K)$ is a finite group.

Then it is claimed that the map $\nu$ induces a group homomorphism $\phi$ from $N$ to the group of affine transformations of $V$ such that for $z\in Z(K)$ and $x\in V$ one has $\phi(z)x=x+\nu (z) $.

I do not understand in which way this function $\phi$ is defined.

how many Q-forms of SL_n(R) are there for a given Q-rank

2012-02-28T16:42:11Z

Let $G$ be a linear algebraic group defined over $\mathbb Q$. Suppose that $G$ is isomorphic to $SL_n$ over $\mathbb R$. Suppose the $\mathbb Q$-rank of $G$ is fixed, say $m$. How many types are there for $G$ up to $\mathbb Q$-isomorphism? Are they finite especially for $m>2$?

proofs of ergodicity of Sl(2, Z) action on R^2 without using duality

2012-08-23T03:18:09Z

The group $ G=SL(2, R)$ acts linearly on $\mathbb R^2$. The Lebesgue measure of $\mathbb R^2$ is invariant and ergodic for $G$. There is a proof using duality theorem: Let $U$ be the upper triangular unipotent subgroup of $G$ and let $\Gamma=SL(2, \mathbb Z)$. Then $U$ acts ergodically on $G/\Gamma$. Therefore duality tells us that $\Gamma$ acts ergodically on $G/U\cong \mathbb R^2\backslash 0$.

This method is good but it can not be extended to the group action on non homogeneous manifolds. Are there any other proofs? It is better that I can see different proofs to learn methods.

The group G^+ of algebraic groups over local fields

2012-07-02T16:24:16Z

Let $G$ be an algebraic group defined over a char 0 local field $k$. Following Borel and Tits (73) we define the group $G^+(k)$ or $G^+$ by the subgroup of $G(k)$ generated by the unipotent elements of $G(k)$.

Suppose $G$ is generated by a finite set of unipotent $k$-subgroups, say $U_1,\cdots, U_n$. Is it true that the group generated by $U_1(k), \cdots, U_n(k)$ is $G^+$?

I feel the answer is positive but do not know how to prove it. It seems that the ideas of the original paper of Borel and Tits can help, but I still do not read French (which I always plan to learn) yet.

surjectivity of rational points induced by surjective map from affine space

2012-06-28T07:07:04Z

Let $k$ be a local field of char $0$ (which is the case I concern). Let $V$ be a variety defined over $k$ and let $f: \mathbb A^n\to V$ be a surjective map (over the algebraic closure of $k$) defined over $k$. Is it true that the restriction of $f$ to $k$ rational points $k^n\to V(k)$ surjective?

After it is answered I realized that I simplified what I want to know too much. Please see the comments for the answer for more information.

Is the image of the representation of the fundamental group associated to a local system discrete?

2012-05-24T14:41:28Z

If $f: X \to S$ is a projective smooth morphism between complex algebraic varieties. Does the $\pi_1(S)$-representation corresponding to the local system $R^i f_* (C_X)$ on $S$ maps $\pi_1(S)$ onto a discrete subgroup of $GL(r, C)$?

unitary representation of semisimple lie groups in view of Moore's ergodicity thm

2010-12-18T15:39:33Z

Let $G=G_1\times\ldots\times G_n$ be a product of (connected) simple Lie groups and $(H, \pi)$ be a unitary representation of $G$. In a proof of Moore's ergodicity thm it uses the following fact $$\pi=\oplus_{I\subset [1,\ldots, n]} \pi_I $$ where $(\pi_I, H_I)$ be the subrepresentation of $\pi$ whose kernel contains $G_i$ for $i\in I$ but no $G_j$ for $j\not \in I$.

How do we prove this fact?

unipotent group and translation invariant metric

2011-05-03T02:44:07Z

Let $U$ be a unipotent upper triangluar group over a local field $K$ of characteristic zero. Can we guarantee that there is a right translation invariant metric on $U$ such that any ball of finite radius is relatively compact?

is the subgroup generated by one-parameter unipotent subgroups a Lie subgroup?

2010-11-24T06:04:21Z

Let $G$ be a Lie group and $H$ be a subgroup generated by some one parameter unipotent subgroups (in group sense). Is it true that $H$ has a Lie group structure which makes it a Lie subgroup of $G$? Is $H$ closed?

How about for general subgroups?

Is restriction of scalars of simply connected algebraic groups still SC?

2011-02-26T06:47:13Z

Let $G$ be a simply connected semisimple algebraic $K$-group and $K$ be a finite extension of $k$. Is $R_{K/k}G$ still a simply connected algebraic group?

We say $G$ is simply connected if for any central isogeny $G'\to G$ is in fact an isomorphism of algebraic groups.

quotient sigma-algebra on quotient space of locally compact groups

2010-12-28T05:40:51Z

Let $G$ be a locally compact second countable topological group and $H$ be a closed subgroup of $G$. Let $\pi: G\to G/H$ be the quotient map. For the Borel $\sigma$-algebra $\mathcal A$ on $G/H$. Is it true that $\pi^{-1}(\mathcal A)$ is the Borel subsets of $G$ consisting of right $H$ orbits?

Is a subset that contains no positive measurable subsets contained in a null measurable set?

2010-12-17T03:27:59Z

Let $(X, \mathcal B, \mu)$ be a "good" measure space, e.g. $\mu$ is a positive Radon measure on a locally compact topological space $X$ with Borel $\sigma$-algebra $\mathcal B$. Let $A\subset X$ such that every measurable subset of $A$ has zero measure. Is it true that there is a zero measure set $B$ such that $A\subset B$?

Is every ball in a Lie group relatively compact?

2010-11-30T15:22:34Z

Let $G$ be a connected Lie group with a left invariant Riemannian metric which induces a metric $d$ on $G$. Does the ball $B_r$ centered at the identity have compact closure?

It is true in easy examples and looks correct.

Comment by ronggang

2013-03-28T08:28:36Z

I see. Thank you.

Comment by ronggang

2013-03-28T08:24:07Z

Krasner's Lemma can be used to prove that every local field of characteristic zero comes from completion of a number field. Therefore it answers the question.

Comment by ronggang

2013-03-28T07:32:09Z

It is known that a local field of characteristic zero is a finite extension of $Q_2$. This why I said that $K*/(K^*)^2$ has order 1,2,4 or 8 is incorrect. The question is whether the order argument is true for local fields coming from completions of number field? If every local field comes from the completion of a number field, then of course the order argument is incorrect.

Comment by ronggang

2012-07-03T02:24:35Z

@Jim: The question is motivated by the p-adic vertion of Rater's Theorem (Margulis Tomanov 94 Invariant measures for actions of unipotent groups over local fields on homogeneous spaces.) They proved the theorem for rational points of unipotent groups, then as a consequence said that the result is also true for groups generated by rational points of unipotent groups. The question arises from my attempt to understand arguments between Theorem 1 and 2 of that paper.

Comment by ronggang

2012-06-29T01:39:26Z

@ Will: It is not surjective even in this case since in general $V^+$ or $V(k)^+$ is not equal to $V(k)$. e.g. PGL_2.

Comment by ronggang

2012-06-28T14:33:33Z

I mean the group generated by the set $f(k^n)$ has finite index in the group $V(k)$. Here is the main example in my mind. Suppose $V$ is generated by connected unipotent groups defined over $k$. Then we get such a map $f$ according to I Porp. 2.2 of Borel's algebraic groups. Then the question is whether the group generated by $k$ points of these unipotent groups has finite index in $V(k)$. If $V$ is semisimple then the question is whether the group they generate contains $V^+$ in the sense of 1973 paper of Borel and Tits( Homomorphismes "Abstraits" de Groupes Algebriques Simples).

Comment by ronggang

2012-06-28T08:24:51Z

Sorry, finite index in $V(k)$.

Comment by ronggang

2012-06-28T08:20:28Z

Thanks. In fact what I want to know is the following: Assume in addition that $V$ is a linear algebraic group. Is it true that the group generated by $f(k^n)$ has finite index in $V$?

Comment by ronggang

2011-02-26T14:06:21Z

I think I would assume $K/k$ is separable. Are there any convenient reference for this full weight lattice characterization?

Comment by ronggang

2010-12-19T01:33:17Z

I think I should give a ref. for this. Ergodic theory and topological dynamics of group actions on homogeneous spaces, M Bekka & M Mayer, Cambridge University Press, P91. In fact, I don't quite believe this fact.

Comment by ronggang

2010-12-01T12:21:24Z

According to Hopf-Rinow, one needs to show the space is complete. This follows from the fact that the metric is left invariant, and it is complete locally!

Comment by ronggang

2010-12-01T00:22:29Z

Thanks, Igor!!!!

Comment by ronggang

2010-11-29T02:12:59Z

Another correction: The above is true in case of two parabolic $k$-subgroups containing a common minimal parabolic $k$-subgroup of ...........;

Comment by ronggang

2010-11-29T02:11:18Z

A correction: $(M_1M_2)(k)=M_1(k)M_2(k)$

Comment by ronggang

2010-11-29T02:10:07Z

Dear Zroslav: This theorem only holds for the rational points over an algebraically closed field. It seems not true that $(M_1M_2)(k)=(M_1M_2)(k)$ which is true in some special cases, e.g. two parabolic $k$-subgroups of a connected reductive $k$-group.