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I was wondering the rationale behind naming parabolic subgroups of linear algebraic groups. The answer, interestingly, comes from the action of SL(2,\mathbb R) on the upper half plane. (I came up with this little discovery on my own.)

The orbit of the point i under the action of the standard parabolic subgroup of SL(2,\mathbb R) is a parabola.

The upper half-plane is an object that comes up in many parts of mathematics, hyperbolic geometry, complex analysis and number theory to name a few. The group SL(2,\mathbb R) acts on it by fractional linear transformations:

\begin{pmatrix} a & b \\ c & d \end{pmatrix} .\ z \mapsto \displaystyle \left( \frac{az+b}{cz+d} \right) .

A parabolic subgroup is it’s subgroup P such that quotienting by P gives a compact variety. Upto conjugation, the only parabolic subgroup of SL(2,\mathbb R) is \begin{pmatrix} * & * \\ 0 & * \end{pmatrix}. It’s action on the point i is given by:

\begin{pmatrix} x & y \\ 0 & x^{-1} \end{pmatrix} . i = \displaystyle \left( \frac{xi+y}{0i+x^{-1}} \right) = y + i x^2

whose locus is a parabola.

I wonder why textbooks in algebraic groups don’t mention this!

EDIT: (21 April, 2015) The above calculation is WRONG. I don’t know the answer to “parabolic”.

Let G = GL_2(F) for some field F and B be the subgroup of upper triangular matrices in G. As usual, let \mathbb P^1 = \mathbb P^1(F) denote the projective plane over F.

As F-varieties, G/B is isomorphic to \mathbb P^1 because B is the stabilizer of the usual (Mobius) action of G on \mathbb P^1. The resulting action of G/B on \mathbb P^1 is 1-transitive and fixed-point free. Indeed, counting points when F is a finite field with q elements agrees with the facts that

\# G(\mathbb F_q) = (q^2-1)(q^2-q)

and

\# \mathbb P^1(\mathbb F_q) = q+1.

This generalizes in that parabolic subgroups (closed subgroup containing a Borel subgroup, i.e., containing the upper triangular matrices upto conjugation, for the general linear group) of an algebraic group give rise to projective varieties. They are precisely the stabilizers of a flag and conversely, the set of flags can be given the structure of an F-space and in particular a topology wherein they are compact spaces (being projective spaces). They are called flag varieties.

In this post, I will give a counter-example, that the image of a hyperspecial maximal compact subgroup under a surjective map need not (even) be maximal. In particular, this will tell us that a maximal compact subgroup (e.g., SO(n) inside SL_n(\mathbb R)) need not be maximal under surjection.

 

Let G be a connected reductive algebraic group over a local non-archimedian field F. A subgroup K of G(F) is called hyperspecial maximal compact subgroup if

  • K is a maximal compact subgroup of G(F),
  • There is a group scheme \mathcal G such that \mathcal G(\mathcal O_F)=K and \mathcal G(\mathcal O_F / \varpi \mathcal O_F) is a connected reductive group over \mathbb F_q, where \varpi is a uniformizer in \mathcal O_F and q is the cardinality of the residue field.

Denote by \phi the canonical surjection

\phi : SL_2 \to PGL_2 = GL_2 / G_m

(over some local field F, or surjection of group schemes, to be pedantic). Choose a non-unit element x of \mathcal O_F. Then

\begin{pmatrix}x & \\ & x^{-1} \end{pmatrix} \not\in SL_2(\mathcal O_F),

but this matrix is in PGL_2(\mathcal O_F) because

\begin{pmatrix} x & \\ &x^{-1} \end{pmatrix} = \begin{pmatrix} x & \\ & x^{-1} \end{pmatrix} . \begin{pmatrix} x & \\ & x \end{pmatrix} = \begin{pmatrix} x^2 & \\ & 1 \end{pmatrix}.

Hence the image \phi(SL_2(O_F) \subsetneq PGL_2(O_F) is compact but not maximal compact (so in particular, not hyperspecial).

 

Remarks:

  • A good reference to read about hyperspecial maximal compact subgroups is Tits’ article in Corvallis (Volume 1). Nick and I are planning to start a reading seminar next semester in Spring 14 on the 40-odd page article.
  • I found this counter-example somewhere on MathOverflow, while I was preparing the Borel-Tate articles (Corvallis, Volume 2) to speak in our “Arthur seminar”.

In my recent number theory seminar on “Hilbert’s 90 and generalizations” (notes here), Professor Goins asked the following interesting question.

Let {K} be a field and {d\in K^*}. Define {T_d} to be the torus

\displaystyle \left\{ \begin{pmatrix} x & dy \\ y & x \end{pmatrix} : x,y \in L, x^2-dy^2=1\right\}.

What values of {d} give {K}-isomorphic tori?

(The question was perhaps motivated by the observation that over the reals, the sign of {d} determines completely whether {T_d} would be split (i.e., isomorphic to {\mathbb R^*}) or anisotropic (i.e., isomorphic to {S^1}).

Here are two ways of looking at the answer.

  • For {d,e \in K^*}, we determine when two matrices {\displaystyle\begin{pmatrix} x & dy \\ y & x \end{pmatrix}} and {\displaystyle\begin{pmatrix} u & ev \\ v & u \end{pmatrix}} are conjugate in {\text{SL}_2(K)}. Solving the system

\displaystyle \begin{pmatrix} x & dy \\ y & x \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} . \begin{pmatrix} u & ev \\ v & u \end{pmatrix} . \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

gives {\displaystyle \frac{e}{d} = \left(\frac{b}{c}\right)^2, de = \left(\frac{d}{a}\right)^2}.

Thus {e \in d.(K^*)^2} i.e., the {T_d}‘s are classified by {\displaystyle \frac{K^*}{(K^*)^2}}. (For {K=\mathbb R}, this is isomorphic to {\{\pm 1\}} so the sign of {d} determines {T_d} upto conjugation.) By Kummer theory, {\displaystyle \frac{K^*}{(K^*)^2} \cong H^1(\text{Gal}(\overline K / K), \mu_2)}, where {\mu_2 = \{\pm 1\}} are the second roots of unity. Thus there is a correspondence between isomorphism classes of tori {T_d \; (d \in K^*)} and quadratic extensions of {K}.

  • Another way to look at the same thing is as follows. Fix {d \in K^*}. Let {L} be an extension of {K} wherein {T_d} splits. Now {T_d(L)} is a split torus of rank 1. For an algebraic group {G} over an algebraically closed field, we have the exact sequence

\displaystyle 1 \rightarrow \text{Inn}(G) \rightarrow \text{Aut}(G) \rightarrow \text{Aut}(\Psi_0(G)) \rightarrow 1,

where {\Psi_0(G)} is the based root datum {(X,\Delta,X\;\check{}, \Delta\,\check{})} associated to {G}. (Here, {X = X^*(G)} and {\Delta} is the set of simple roots of {X} corresponding to a choice of a Borel subgroup of {G}.) For details, see Corollary 2.14 of Springer’s paper “Reductive Groups” in Corvallis.

In our case, {G = T_d} so {\Psi_0(G) = ( \mathbb Z, \emptyset, \mathbb Z \;\check{}, \emptyset)}.

\displaystyle \text{Aut}(\Psi_0(G)) \cong \text{Aut}(\mathbb Z) \cong \{ \pm 1\}.

Now {L/K} forms of {T_d} are in bijective correspondence with

\displaystyle H^1(\text{Gal}(L/K), \text{Aut}(\Psi_0(G))) = H^1(\text{Gal}(L/K), \{\pm 1\}) \cong \text{Hom}_{\mathbb Z}(\text{Gal}(L/K), \{\pm 1\});

the last isomorphism because the Galois group acts trivially on the split torus {T_d(L)}. {\blacksquare}

Recently (22 March 2013) I took my advanced topics exam. In spirit of the Princeton Generals, I wrote out my interview questions for use to anyone in these special topics.

Topics: linear algebraic groups, class field theory

Committee: Freydoon Shahidi (chair), David Goldberg

Linear algebraic groups:

Goldberg decided to start with Linear algebraic groups. What is meant by ‘split’? (I told I had prepared only the algebraically closed case. They were okay). What is parabolic? If Q is parabolic in G and P is parabolic in Q, prove that P is parabolic in G. (Went totally blank. They gave hints – use the other equivalent definition).

State Bruhat decomposition. (I started defining terms – maximal torus, root system, Weyl group etc.) How is the Weyl group related to the torus? (Another moment when I blanked out). How does W act on T? (By conjugation. So W is the quotient of the normalizer of the maximal torus by its centralizer). When is the centralizer equal the torus? (When it is maximal). Shahidi objected and corrected me twice when I wrongly pronounced “veil” for Weyl group – correct is “vile”. There is another important group after Andre Weil.

Bruhat decomposition for GL_n. What is it’s Weyl group? (S_n). What is the length of an element? (minimal length of the decomposition in terms of reflections). What is the polynomial the big cell satisfies? (I had prepared this one. Told the answer and that the proof goes by induction but Shahidi was not satisfied. He told something I didn’t quite understand. I gave the decomposition explicitly and he told it generalizes the well-known LU decomposition to classical groups).

What is the critical step in the classification of semisimple groups of rank 1? (Uses Bruhat decomposition). Give a sketch of proof. (I couldn’t. But turns out, they really wanted me to state the theorem. SL_2 and PSL_2).

Given that the normalizer of a parabolic is itself, show that it must be connected. (Missed a step in that one can choose the conjugating element inside the connected component).

[ That was all about algebraic groups. Nothing about Dynkin diagrams, finding parabolics, simple connectedness and fundamental groups of classical groups. ]

————————————————————————————————————-

Number Theory:

Class group of Q(\sqrt -7). (Minkowski bound works).

Class group of Q(\sqrt 21). (I saw that 25 – 21 = 4 and started considering the factorization of the prime 2 above using quadratic reciprocity but got it wrong. It took me almost 10 grueling minutes to figure out that 21 was not a prime! Embarrassment).

Define the Hilbert symbol. State the norm condition. Prove the bi-multiplicativity property. (Consequence of a group homomorphism). State and prove Hilbert’s reciprocity. (I had prepared a classical proof from Serre but he insisted on proving via Class field theory. Proved that the local p-Artin maps glue to give a global Artin map and the theorem follows elegantly).

Goldberg asked if I knew anything about the generalized symbol. (I stated and mentioned the skew-symmetry property). He said it’s not in your syllabus anyways.

Show that p = a^2 + ab + b^2 precisely when p is 1 (mod 3). (I was looking at the field of cube roots of unity but was blundering some norm calculation. After many frustrating minutes for everyone, it was discovered that the problem was wrong and should have p = a^2 – ab + b^2. I commented, “I have been doing this calculation since tenth grade” on which Prof. Shahidi quickly retorted, “still you haven’t memorized it!”

—————————————————————————————————————

The exam lasted for 90 minutes. They told me to wait outside and after a stressful 5-minute period for me, they came out and congratulated me.

Shahidi: Good, congratulations.

Me: Thanks.

Shahidi: Don’t see me for a month now.

Me: Yeah, the past two weeks were tough on me too.

Shahidi: Let us not meet each other for a while now. (All smile in acknowledgement).

Today’s post is about group theory. Well, to state the result, one needs to know just group theory but proving it requires some knowledge of root systems. And one really understands its application when studying algebraic groups. In my opinion, one can rarely say while learning mathematics that s/he will not need a particular result, or something is not useful in their research. So it’s always good to know something and connect to it later when learning something related.

The Bruhat decomposition is a decomposition of a group having a “Tits system” into double cosets. It is used to decompose GL(n,F), the group of n \times n invertible matrices over a field F, or any reductive connected closed subgroup of this matrix group.

For a group G, a Tits system (named after Jacques Tits) is a triple (B, N, I), \, B, N subgroups of G called a BN-pair and I is a set such that the following axioms hold:

  1. T := B \cap N \unlhd N,
  2. I is a set of generators of W := N / T (called the Weyl group) and s^2 = 1 for all s \in I,
  3. If w \in W and s \in I, then wBs \subseteq BwsB \cup BwB,
  4. If s \in I, sBs^{-1} \neq B,
  5. B, N generate G.

Remarks:

  • (1) says that W is a group.
  • In (3), $W$ is not necessarily a subgroup of G. Yet, define wB as wB := nB, where $w=nT$ and observe the well-definedness.
  • We’ll be concerned with “Bruhat cells”, C(w) := BwB. With this notation, (3) is equivalent to C(w) C(s) \subseteq C(ws) \cup C(w).
  • From (4), 1 \not\in I.
  • From (2), each s \in I has order 2.

Theorem (Bruhat decomposition). 

Let (B, N, I) be a Tits system and W = N/T be the Weyl group. Then,

G = \coprod_{w\in W} C(w).

A (rough) sketch of the proof:

Let G' = \cap_{w\in W} C(w). Prove that $G’$ is a group, and in fact a subgroup of G.

G' contains N and B which generate G so G' = G.

For disjointness, it suffices to show w_1 = w_2 whenever C(w_1) = C(w_2).

Observe that two double cosets are either disjoint or equal.

The canonical example:

F is any field and GL(n,F) is the set of invertible matrices over F. The Borel subgroup B consists of upper triangular matrices and take N to be the subgroup of monomial matrices, i.e., each row and each column has a unique nonzero entry. Then T = B \cap N \cong (F^*)^n, which is a torus. One does however need to show that this forms a Tits system.

Recently while browsing in the Purdue Math library, I found notes of Armand Borel‘s talk titled `Algebraic groups and Arithmetic groups’. It gives an overview of algebraic groups and its applications. They are very sleek (but mathematically dense) so I decided to type them out. These notes discuss Tits systems, and their deep connections to Geometry, group theory and number theory. You can read them here.

About me

Abhishek Parab

I? An Indian. A mathematics student. A former engineer. A rubik's cube addict. A nature photographer. A Pink Floyd fan. An ardent lover of Chess & Counter-Strike.

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