Closures of certain matrix varieties and applications

We prove some results about closures of certain matrix varieties consisting of elements with the same centralizer dimension. This generalizes a result of Dixmier and has applications to topological generation of simple algebraic groups.


Introduction
Let k be an algebraically closed field of characteristic p 0 and let M n (k) denote the algebra of n × n matrices over k.If A ∈ M n (k), we let D(A) denote the subset of M n (k) consisting of all elements with the same Jordan canonical form as A up to changing the eigenvalues (but with the same number of distinct eigenvalues).
Note that D(A) is closed under conjugation by GL n (k) and any two elements in D(A) have conjugate centralizers.More generally, the fixed space of any two elements in D(A) on the dth Grassmanian, G d are also conjugate and in particular the dimension of the fixed space on G d is constant on D(A).
We generalize a result of Dixmier [2] (for the case of type A).Let G be a simple algebraic group over an algebraically closed field.A unipotent element u ∈ G is called a Richardson element if there exists a parabolic subgroup P such that u is in the unipotent radical of P and the P-conjugacy of u is an open dense subset of the unipotent radical of P. One similarly defines Richardson nilpotent elements in the Lie algebra of G. Diximier proved (in characteristic 0) that if A is Richardson, then A is the limit of semisimple elements such that the dimensions of the centralizers of each of the semisimple element are the same as the dimension of the centralizer of A. This was used by Richardson [9] to prove that the commuting variety of a reductive Lie algebra g in characteristic 0 is an irreducible variety of dimension equal to dim g + rank(g) (this was proved for gl n by Motzkin and Taussky [8] in all characteristics).Levy [6] observed that Dixmier's result goes through in good characteristic as well and so Richardson's proof for the irreducibility of the variety of commuting pairs goes through in good characteristic as well.
We generalize Dixmier's result for gl n by obtaining the same conclusion in arbitrary characteristic (using the Zariski topology) and for arbitrary elements.We also prove a sandwich result by showing that any such element is trapped between semisimple and equipotent elements (i.e.having a single eigenvalue) all with the same centralizer dimension.We also need to verify that our semisimple and nilpotent elements have some extra properties (required for applications).We combine this with a recent result of Guralnick and Lawther [5,Proposition 3.2.1] to conclude that all the fixed point spaces on all Grassmanians are the same for elements in these subvarieties.
This will be used in [4] to deduce some results related to topological generation of simple algebraic groups and to extend the results of [1] from semisimple and unipotent classes to all classes.
Our main result is the following: Then there exist a semisimple element S ∈ M n (k) and an equipotent element N ∈ M n (k) such that the following hold: (

2) The Zariski closure of D(A) contains D(N ). (3) A, S and N all have centralizers of the same dimension. (4) If 1 d < n, then A, S and N all have fixed point spaces of the same dimension on G d .
Note that in our proof controlling the determinant or trace of the elements is not an issue and so the same results hold for sl n (k), GL n (k) and SL n (k).We make some remarks in the last section regarding the symplectic and orthogonal cases.
We also give some consequences regarding generation of simple algebraic groups that will be required in the sequel [4].(3) was observed in [3] for the special case when A is semisimple (i.e. the existence of a nilpotent class with the same centralizer dimension and with the largest eigenspace of the same dimension) and was used to prove results about generic stabilizers.We give the proof in the next section and some applications in the following section.

Proofs
If A ∈ M n (k) has m distinct eigenvalues, we set (A) to be the set of m partitions i where i is the partition associated to the Jordan blocks of each generalized eigenspace of A.
Let X ( ) = {A ∈ M n (k) | (A) = } for a set of m partitions whose sizes add up to n.Note that if A, B ∈ X ( ), then the centralizers of A and B are conjugate and in particular have the same dimension.Let = i where by the sum of partitons we mean the usual addition (just view a partition as row vector with nonincreasing entries-adding 0's to make the vectors have the same length).Given a partition , let be the transpose partition.
If is a partition of n, let U ( ) be set of all matrices with a single eigenvalue and the sizes of the Jordan blocks be given by and let S( ) be the set of semisimple matrices with the dimensions of the eigenspaces given by .
We first note the following elementary result.

Lemma 2.1
Let A be an upper triangular matrix with diagonal entries contained in the set {a 1 , . . ., a s } with the multiplicity of a i equal to d i .Assume that the entries in the i, i + 1 positions are all nonzero.Then the Jordan canonical form for A consists of one Jordan block for each a i and it has size d i .
Proof Note that A is regular (i.e. its centralizer has dimension n).This follows by noting A is cyclic (i.e. the column vector e n = (0, 0, . . ., 1) generates the module of column vectors for the algebra k[A]).Thus, the characteristic and minimal polynomials of A are both s i=1 (x − a i ) d i and the result follows.
Let A(a 1 , . . ., a s ; e 1 , . . ., e s ) denote the matrix above with the entries i, i + 1 all equal to 1 and all other entries (besides the diagonal) 0. Consider the s-dimensional affine space of all such matrices (as the a i range over all possibilities).Note that the generic points in the variety of all such matrices (i.e. with a i = a j for i = j) are all regular.If all the a i = a, then the matrix is a regular equipotent matrix.
If A ∈ M n (k), let G A d be the fixed space of A on G d (i.e. the set of d-dimensional subspaces W of k n with AW ⊆ W ).
Our first result is the following: Theorem 2.2 Let be as above.Let = .
Let m 1 m 2 • • • m s be the pieces of the partition of .An element of S( ) has m 1 distinct eigenvalues and more generally has m j distinct eigenvalues having multiplicity at least j.Set d = m 1 .
For any a 1 , . ., a d ∈ k, let B be the matrix with diagonal blocks of size m i with the ith diagonal block being A(a 1 , . . ., a m i , 1, . . ., 1).
Note that if the a i are distinct, then B is semisimple and is in S( ) and this is a generic point in the affine space of dimension m obtained by allowing all possibilities for a i .In the closure are the elements when all a i are equal and so X ( ) is in the closure of S( ) as claimed.
In the general case, we just choose a B j as above corresponding to the Jordan blocks corresponding to the jth eigenvalue of an element in X ( ) giving another copy of affine space.The result follows by considering each block separately.
We prove (2) similarly.Let A ∈ X ( ).First consider the case that each partition in has just one part.By taking generic elements and taking elements in the closure with a single eigenvalue, we see we can obtain a regular element with a single Jordan block (and any eigenvalue).In general, we decompose A into pieces corresponding to = with the jth piece having a single Jordan block corresponding to each eigenvalue of size the jth part of and the closure contains elements with a single eigenvalue with the partition of Jordan blocks corresponding to .
The formulas for the dimension of centralizers and semisimple elements and nilpotent elements give that the centralizers have the dimension for elements in S( ) and U ( ).This observation together with (1) and ( 2) proves (3).
The equality of the dimension of the fixed spaces on Grassmanian follows for the unipotent and semisimple elements by [5,Proposition 3.2.1].Then by ( 1) and ( 2), (4) follows.
Clearly, the result (with essentially the same proof) holds with M n (k) replaced by either GL n (k) or SL n (k).

An application
We now apply Theorem 2.2 to the action on Grassmanians.Recall that A ∈ M n (k) fixes a subspace W means that AW ⊆ W .
Let d(A) be the dimension of the largest eigenspace of A. Observe that if = with = (A), then d(A) = d(B) = d(C) by Theorem 2.2 applied to G 1 (or by observation).
We now generalize the result [1, Lemma 3.35] which was stated for unipotent or semisimple elements.
Then one of the following holds: (1) Each A i has a quadratic minimal polynomial and s = 2; or (2) i dim Proof There is no harm is adding a scalar to each A i and so we may assume that the A i are all invertible.This is proved in [1, Lemma There should be an analogous but more complicated result both for higher dimensional Grassmanians and also for actions on the variety of totally singular spaces of a given dimension.
We note that one can generalize Dixmier's result for arbitrary elements in the symplectic and orthogonal Lie algebras in characteristic not 2. Indeed, any element that does not have 0 as an eigenvalue is in the Zariski closure of the set of semisimple elements with the same centralizer dimension.There is a similar statement for the groups.In good characteristic, Diximier's result still holds for Richardson nilpotent and unipotent elements.We do not require this for our application (even for groups of this type) since these groups have dense open orbits on the Grassmanians (indeed, they have only finitely many orbits on each Grassmanian) and so the result for gl is sufficient.
3.35] in the case each A i is either semisimple or unipotent.The previous result shows that this implies the result for arbitrary A i .Let H be a closed irreducible subgroup of GL n (k) that has a dense orbit O on G 2 .Let x 1 , . . ., x s ∈ H with d(x i ) (s − 1)n.Assume moreover that either s > 2 or s = 2 and x 1 does not have a quadratic minimal polynomial.Then for generic h i ∈ H, x h 1 1 , . . ., x h s Proof It follows by the previous result that Theorem 3.1 (2) holds.Let C i be the Hconjugacy class of x i .By considering H acting on O, it follows by [1, Lemma 3.14] that the subset of C 1 × C 2 × • • • ×C s that has a fixed point on O is contained in a proper closed subvariety of C 1 × C 2 × • • • ×C s .Note that this result holds for H a symplectic or special orthogonal group acting on nondegenerate 2-spaces.
s do not fix a point of O.