User turtle - MathOverflow

Zariski density of conjugates of subgroups by arithmetic subgroups?

2011-03-14T16:55:27Z

Let $G$ be a linear algebraic $\mathbb{Q}$-group, which is assumed to be connected, $\mathbb{Q}$-simple, and of adjoint type, such that the Lie group $G(\mathbb{R})$ has no compact factor defined over $\mathbb{Q}$. Let $\Gamma\subset G(\mathbb{Q})$ be a congruence subgroup. It is known, from the theory of Margulis, that $\Gamma\subset G(\mathbb{R})$ is Zariski dense. For convenience assume that $\Gamma\subset G(\mathbb{R})^+\cap G(\mathbb{Q})$ and that $\Gamma$ is torsion free. Note also that in this case, if one takes $X$ to be the non-compact symmetric domain associated to $G(\mathbb{R})^+$, then the quotient $X/\Gamma$ is a localy symmetric manifold of negative curvature (a typical example of hyperbolic manifold}.

I'd like to consider conjugates of linear $\mathbb{Q}$-subgroups of $G$ under $\Gamma$. More restrictively, let me take $H\subset G$ a connected semi-simple $\mathbb{Q}$-group such that $H(\mathbb{R})$ again has no compact factors defined over $\mathbb{Q}$. Then

(1) is the union $\bigcup_{g\in\Gamma}gHg^{-1}$ Zariski dense in $G$?

(2) if $\Gamma'$ is a finitely generated subgroup of $\Gamma$, and $H'$ be the Zariski closure of the subgroup of $G(\mathbb{Q})$ generated by $\bigcup_{g\in \Gamma'}gH(\mathbb{Q}) g^{-1}$, then how far is $\Gamma'$ from being an arithmetic subgroup of $H'$?

Thanks!

symplectic representations: when could the center act trivially?

2011-11-30T12:01:26Z

I'm considering a problem about symplectic representation of real reductive group, which fits more or less into the setting of symplectic representations discussed in Milne's survey ''Shimura varieties and moduli''.

My question is as follows: let $G$ be a connected reductive algebraic group over $\mathbb{R}$, $\rho:G\rightarrow GSp(V)$ a faithful algebraic representation on a finite-dimensional real vector space $V$, preserving a symplectic form $\psi$ on $V$ up to scalars. Assume that there exists a homomorphism $h:\mathbb{S}\rightarrow G$, $\mathbb{S}=Res_{\mathbb{C}/\mathbb{R}}\mathbb{G}_\mathrm{m}$, such that $\rho\circ h$ is a Hodge structure of type ${(-1,0),(0,-1)}$, namely a complex structure on $V$, and such that $\psi$ serves as a polarization of this complex structure. If we know further that the action of $G^{der}$ the derived group of $G$ on $V$ does not admit trivial subrepresentation of dimension $>0$, can we deduce that the connected center $C$ of $G$ acts on $V$ through a split torus over $\mathbb{R}$?

Note that if $G$ stabilizes a real subrepresentation $V'$ of $V$ on which $G^{der}$ acts trivially, then the Hodge structure on $V'$ is determined by the action of the connected center $C$ of $G$, and thus $C$ could not split over $\mathbb{R}$, otherwise $V'$ would be a sum of Tate twists. So my question is put in the inverse direction: if $G^{der}$ acts without trivial subrepresentation, does this force $C$ to be split?

An easy case to be considered is when $V$ is a simple real representation of $G$, which is given as $V=M\otimes N$ for $M$ a simple representation of $G^{der}$ and $N$ a simple representation of $C$. Here I pass from $\mathbb{S}$ to the action of Lie algebra $\mathbb{C}\rightarrow\mathfrak{g}^{der}\oplus\mathfrak{c}$, with $\mathfrak{g}$ the Lie algebra of $G$, and $\mathfrak{c}$ the Lie algebra of $C$. Since there are only two indices appearing in the Hodge type of $M\otimes N$, we deduce that one of $M$ and $N$ is of a single Hodge type, hence a Tate object, and the other is of Hodge type ${(r-1,r),(r,r-1)}$. And one must have $N$ to be of type $(-r,-r)$ by the assumption of $G^{der}$, which shows that $C$ acts on $V=M\otimes N$ through a split torus.

The case I fail to attack in this way is when $V$ is a subrepresentation of a representation of the form $M\otimes N$, where $M$ is a simple representation of $G^{der}$ and $N$ a simple representation of $C$. By Bourbaki Algebra Chap.8 Sect.7, we know that finite-dimensional simple modules of $\mathfrak{g}=\mathfrak{g}^{der}\oplus\mathfrak{c}$ does arise in this way. However in this case my arguments via Hodge types do not work, and it seems possible that $M\otimes N$ admits a non-trivial simple subrepresentation.

And thus I would like to know if there are other ingredients I'm missing for the split center $C$, or perhaps there are counter examples? I also expect the same situation happen for the rational case, namely $G$ comes from some Shimura subdatum in the Siegel datum $(GSp(V),H(V))$, and the claim on its connected center $C$ is modified as: $C$ splits over a totally real field.

thanks a great deal for reading my lengthy descriptions.

reciprocity maps and norm maps in class field theory

2011-11-04T12:32:50Z

we consider number fields embedded in a fixed algebraic closure $k$ of $\mathbb{Q}$, or simply consider number fields in $\mathbb{C}$. for $L$ a number field, we write $G_L$ for the galois group $Gal(k/L)$.

Take $F$, $F_1$, $F_2$ and $E$ number fields such that $F=F_1\cap F_2$, $E=F_1F_2$, and $F_1$ Galois over $F$. Then $G_E=G_{F_1}\cap G_{F_2}$ and $G_F=G_{F_1}G_{F_2}$ (as we have assumed that $F_1$ is galois over $F$).

Then what happens with the abelianization? the norm maps give us a diagram

$$\xymatrix{ G_E^{ab} \ar[d]^{\subset} \ar[r]^{\subset} & G_{F_1}^{ab} \ar[d]^\subset \ G_{F_2}^{ab} \ar[r]^\subset & G_F^{ab}}$$

Is there chance that this diagram is cartesian?

We can also write $T^L$ for the torus $Res_{L/\mathbb{Q}}\mathbb{G}_\mathrm{m}$. Then the diagram above can be produced from the one below by passing to connected components of idele classes

$$\xymatrix{T^E \ar[d] \ar[r] & T^{F_1} \ar[d] \ T^{F_2} \ar[r] & \T^F}$$

However by dimension arguments one cannot expect this lower diagram to be cartesian. What happens if we only take connected components of the idele class groups?

thanks.

a question about invariant volume forms on homogeneous spaces.

2011-10-07T15:21:15Z

Here I consider $G$ a connected Lie group, which is assumed to be linear (i.e. embeddable in some $GL_n(\mathbb{R})$, and $X$ a homogeneous space under $G$. Fix a point $x\in X$, one considers the map $m:G\rightarrow X$ sending $g$ to $g(x)$.

Does the left (or right) invariant volume form on $G$ passes to an invariant volume form on $X$, under the pushing forward along $m$? Here by pushing forward along $m$, I mean the measure $\mu$ on $X$, such that for a continuous function $f$ of compact support, one has $$\int_X f(x)d\mu(x):=\int_G f(m(g))dg$$, $dg$ being the left (or right) Haar measure on $G$.

It seems that one needs to assume that the isotropy subgroup of $x$ in $G$ is compact. Does it matter if $G$ is not unimodular?

Many thanks.

a naive question about p-adic local monodromy theorem

2011-08-25T00:00:48Z

The question is about whether one can view the p-adic local monodromy theorem as the quasi-unipotence of some monodromy operator.

it is known that the classical local monodromy theorem (i.e. for variation of Hodge structures) can be stated in terms of "linear algebra": let $\mathbb{V}$ be a polarized variation of Hodge structures on the punctured disc $\Delta^\times$. Then the monodromy operator $T$ is quasi-unipotent. Here the monodromy means the representation of the topological fundamental group on any fiber $V$ of $\mathbb{V}$, i.e. $\rho:\pi_1(\Delta^\times)\rightarrow GL(V)$, which is determined by the underlying local system $\mathbb{V}$. And this representation is simply characterized by $T=\rho(1)$ as one may identify $\pi_1(\Delta^\times)$ with $\mathbb{Z}$.

In the p-adic setting one has the Crew conjecture, also known as the p-adic local monodromy theorem, proved by Andre, Mebkhout, and Kedlaya. It can be stated as: let $F$ be the fraction field of a Witt ring of perfect residue of char.p, $R$ the Robba ring over $F$ (a ring of Lauent series convergent on some "thin" open annalus of outer radius 1 defined over $F$), and $M$ a differential module over $R$ with a compatible Frobenius structure (a slope zero $\phi$-module for the Frobenius structure on $R$ coming from the standard one on the Witt ring), then $M$ is quasi-unipotent, in the sense that after a finite extension of $R$ to some other Robba ring (coming from a finite extension of $F$), $M$ becomes unipotent, i.e. a successive extension of the trivial differential module $(R,d)$.

My question is whether one can find out a similar interpretation of quasi-unipotence. Namely, let $M$ be such a differential module with Frobenius structure on $R$, then show that it gives rise to a quasi-unipotent representation of the fundamental group of the annalus on a suitable space, possibly the solution space of $M$?

how large can this pro-p quotient be?

2011-03-03T08:21:21Z

Let $p$ and $\ell$ be distinct rational primes. Note that the unit group of the finite field $\mathbb{F}_\ell$

is of order $\ell-1$, hence there is the probability of finding a $p$-quotient from $\mathbb{F}_\ell^\times$.

When taking the infinite product $\prod_{\ell\neq p}\mathbb{F}_\ell^\times$, how large could its maximal pro-p quotient be? And do we have a comuptable bound for the p-valuation of $\ell -1$ when $\ell$ tends to infinity?

(My original question was about pro-p quotient of the tame inertia group of a p-adic local field, which didn't make sense for obvious reason (pointed out as below), and I propose the replace it with this one)

intersection of a parabolic subgroup with a subgroup

2010-12-27T15:42:11Z

I'm interested in the following question: let $k$ be a field of characteristic zero (just for simplicity), $G$ a connected semi-simple $k$-group, $P\subsetneq G$ a parabolic $k$-subgroup, and $H\subsetneq G$ a connected $k$-subgroup. Assume that $H\nsubseteq P$, then does the intersection $H\cap P$ becomes a parabolic $k$-subgroup for $H$? One might even assume that $H$ is reductive.

decomposition into irreducible unitary representations: references for explicit formulas?

2010-11-07T13:24:43Z

I'm looking for references of the decomposition of $L^2(\Gamma\backslash G)$, where $G$ is a connected Lie group, and $\Gamma\subset G$ a discrete lattice; for simplicity one may assume that $G$ is the real point associated to a linear algebraic group defined over $\mathbb{Q}$, without characters defined over $\mathbb{Q}$, and $\Gamma$ is an arithmetic/congruence lattice in $G$. Write $\Omega=\Gamma\backslash G$. Then $\Omega$ has a canonical probability measure induced by the left Haar measure of $G$, and the right translation gives a unitary representation on $L^2(\Omega)$. When I consider the decomposition of $L^2(\Omega)$ into irreducible unitary representaions, I heard about the notion of continuous spectra and discrete spectra, but why are they called spectra and where may I find explicit descriptions for the classical groups?

Also what if one considers the double quotient $M=\Gamma\backslash G/K$, where $K$ is a maximal compact subgroup of $G$? Is it also described via representations of $G$? I don't see an explicit action of $G$ on it. Also if I conjugate $K$ to a second maximal compact subgroup, is there any invariant description of the decomposition of $L^2(M)$, namely independent of the choice of $K$? I heard about the notion of Shimura varieties, so maybe here I should restrict to the case where $M$ is a locally symmetric hermitian manifold.

Thanks a lot!

unipotent groups, their forms and representations

2010-10-04T17:50:08Z

For simplicity fix a base field $k$ of characteristic zero, and consider smooth affine algebraic $k$-groups. (It is understood that unipotent groups in positive characteristic are more complicated, as one might have interesting non-smooth ones.)

Question 1: forms of unipotent groups

If $k$ is algebraically closed, then it is clear that every connected unipotent $k$-group is a successive extension of $\mathbb{G}_\mathrm{a}$'s. Then what about the case $k$ not algebraically closed? Is there non-trivial $K$-forms of, say, the upper trangular unipotent $k$-group of $GL_n$, and of the unipotent radicals of Levi $k$-subgroups of $GL_n$, etc?

If $K$ is a finite Galois extension of $k$ of Galois group $\Gamma$, then a $K$-form of the split $k$-torus is the same as a $\Gamma$-module structure on the group of characters $\mathbb{Z}^d$. Is there analogous results for $K$-forms of unipotent $k$-groups?

For example, with $K$ a finite extension field of $k$. the scalar restriction $$Res_{K/k}\mathbb{G}_{\mathrm{m}}$$

is not split as $k$-torus, but $$Res_{K/k}\mathbb{G}_\mathrm{a}$$

splits into a direct sum of $\mathbb{G}_\mathrm{a}$, becasue $K$ is a finite vector space over $k$ viewed additively. It is from this example that I want to know if there are interesting examples of forms of unipotent groups.

Question 2: representations of unipotent groups

If one has the one dimensional unipotent group $U=\mathbb{G}_\mathrm{a}$, then an algebraic representation of $U$ on $V$ a finite dimensional $k$-vector space is the same as a unipotent operator on $V$. One can then extend this description naturally to obtain the Tannakian category of finite dimensional algebraic representations of $U$.

And what about general unipotent $k$-group $U$? By the theorem of Lie-Engel, we know that such a representation of $U$ on $V$ is upper-triangular: it stabilizes a full flag of $V$, and acts trivially on the successive quotients (because of unipotence). Is there more precise information one can find about these representations so as the determine the Tannakian category of representations of $U$?

Again, let $K$ be a finite Galois extension of $k$ with Galois group $\Gamma$, and $U$, $W$ two connected unipotent $k$-groups that are isomorphic over $K$. Then how can one distinghuish the representations of the two groups by some "action" of $\Gamma$ one the representations spaces, in the spirit one finds in representations of the $k$-torus $$Res_{K/k}\mathbb{G}_\mathrm{m}$$

thanks!

Centralizers and Cartan involutions

2010-10-04T11:35:42Z

This should be an easy question about centralizers in reductive lie groups, but I wonder if it is already available from the literature.

Consider $G$ a connected non-compact semi-simple Lie group, with a Cartan involution $\sigma$, and $H$ a reductive subgroup, stablized by $\sigma$. Then the fixed part of $\sigma$ in $H$ is a maximal compact subgroup $K_H$ in $H$. Compare the two centralizers $Z(K_H,G)$ and $Z(H,G)$. Are they equal, or at least, their derived parts equal to each other?

Thanks.

Comment by turtle

2011-08-25T00:07:12Z

it is understood that such differential module with compatible Frobenius structure is equivalent to a p-adic Galois representation, as is the starting point of the theory of $\phi,\Gamma$-modules of Fontaine, Colmez, etc. In that way does the unipotence of the differential modules implies any unipotence on the p-adic Galois representation?

Comment by turtle

2011-03-15T07:25:42Z

thanks! Corrected as suggested.

Comment by turtle

2011-02-17T08:54:06Z

thanks a lot! and I'm sorry to have posted such a stupid question. I vote to delete this item.

Comment by turtle

2011-01-05T13:40:33Z

I'm confused. Say $\mathfrak{g}$ is the Lie algebra of a compact Lie group $G$, and $\mathfrak{t}$ is a 1-dimensional semi-simple Lie subalgebra, which is the Lie algebra of a 1-dimensional compact torus $T$ in $G$. Via the adjoint representation one also gets a $\mathbb{Z}$-grading on $\mathfrak{g}$, because the irreducible representations of $T$ is also parameterized by $\mathbb{Z}$. The grading is also compatible with the Lie bracket in the above sense, but the non-negative part does not corresponds to a parabolic subgroup of $G$, as $G$ is compact. Something missed in the statement?

Comment by turtle

2010-11-11T08:11:55Z

it seems that the elliptic surface is fibered over the modular curve by compact tori, and thus the new contribution to the spectrum decomposition should appear only in the discrete part, although I know nothing about the concrete formula. And for the Kuga varieties over compact Shimura curves, then still there is only a discrete spectrum.

Comment by turtle

2010-11-08T14:23:06Z

I've started with two books of Iwaniec on modular forms, mainly the chapters on spectral decomposition. I haven't plunged into the volumes of Helgason yet, and I don't know if I should find detailed presentations of the locally symmetric manifolds there. The second example I'm considering is a non-reductive group: $\mathbb{R}^2\rtimes GL_2(\mathbb{R})$. This group can produce univrsal elliptic curves, also called elliptic surfaces. I want to now check the spectral decomposition for the universal elliptic curve. I'm quite lost for literatures in this topic, or maybe I'd try some Jacobi forms.

Comment by turtle

2010-10-05T12:08:23Z

@BCnrd: thanks so much for the reply and comments. I'm really sorry about overlooking these starting examples. It seems that reductive subgroups not of maximal ranks are complicated, as I have also found from similar discussions in mathoverflow. Thanks again, and I'd call a stop for this question.

Comment by turtle

2010-10-04T16:33:28Z

thanks for the counter example. Does anything change if $H$ is required to be conencted and semi-simple?