Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mccme.ru/~panyush/law08-short.pdf Äàòà èçìåíåíèÿ: Sat May 31 01:58:12 2008 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 00:32:41 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: http astrokuban.info astrokuban

On the irreducibility of commuting varieties
D. Panyushev
Independent University of Moscow Russia

30.05.2008 / LAW '08

D. Panyushev (Moscow)

On the irreducibility of commuting varieties

LAW '08

1 / 21


1

Reductive Lie algebras and commuting varieties Matrix commuting varieties Richardson's theorem Generalisations Triples of matrices Natural non-reductive subalgebras of gln Nilpotent commuting varieties Knutson's diagonal scheme Involutions and commuting varieties Some open problems

2

3 4

Main questions: Is a commuting variety irreducible? If not, what are the irreducible components?
D. Panyushev (Moscow) On the irreducibility of commuting varieties LAW '08 2 / 21


Reductive Lie algebras and commuting varieties

Matrix varieties

Pairs of commuting matrices
The ground field is C.

Definition
C (2, n) = {(A, B ) | AB = BA} gln â gln

Theorem
C (2, n) is irreducible for any n. T. S. M OT Z K I N and O L G A TAU S S K Y (1955) M . G E R S T E N H A B E R (1961) R . W. R I C H A R D S O N (1979) Y U. N E R E T I N (1987) R . G U R A L N I C K (1992)

D. Panyushev (Moscow)

On the irreducibility of commuting varieties

LAW '08

3 / 21


Reductive Lie algebras and commuting varieties

Matrix varieties

Neretin's proof of irreducibility
One has to prove that (A, B ) C (2, n) can simultaneously be diagonalised after a "small" deformation. W,µ = ker (A - I )n ker (B - µI )n ; Replace Cn with W,µ and assume that A, B are nilpotent; (The argument is by induction on max dim W,µ ). p, q C s.t. R := pA + qB is non-regular. Then a semisimple C such that tr(C ) = 0 and [C , R ] = 0. Deformation in W,µ : (A, B ) (A - qC , B + pC ), C. By the induction assumption, (A - qC , B + pC ) can be approximated by commuting semisimple matrices.

D. Panyushev (Moscow)

On the irreducibility of commuting varieties

LAW '08

4 / 21


Reductive Lie algebras and commuting varieties

Richardson's theorem

H is a connected algebraic group, h = Lie (H ). For x h, zh (x ) is the centraliser of x in h.

Definition
The commuting variety of a Lie algebra h is C (h) = {(x , y ) h â h | [x , y ] = 0}. Example: C (2, n) = C (gln ). If a h is commutative, then H ·(a â a) C (h). The very first step: Consider the projection C (h) h and look at the dimension of fibres. One readily obtains that dim C (h) dim h + min dim zh (x ) = 2 dim h - max dim{H-orbits in h}.

D. Panyushev (Moscow)

On the irreducibility of commuting varieties

LAW '08

5 / 21


Reductive Lie algebras and commuting varieties

Richardson's theorem

Richardson's theorem­1
Theorem (R.W. Richardson, 1979)
If g is reductive, then C (g) is irreducible. More precisely, if t is a Car tan subalgebra, then C (g) = G·(t â t). Plan of proof
1

We have to approximate any (x , y ) C (g) by a pair of commuting semisimple elements. The Jordan decomposition and induction on rk [g, g] allow us to assume that x , y are nilpotent. Next, one can assume that zg (x ) contains no semisimple elements. (Such an x is called distinguished.) For x distinguished, one uses some proper ties of sl2 -triples.

2

3

4

D. Panyushev (Moscow)

On the irreducibility of commuting varieties

LAW '08

6 / 21


Reductive Lie algebras and commuting varieties

Richardson's theorem

Richardson's theorem­2
Suppose x is distinguished and {x , h, y } is an sl2 -triple. Then x + y G h for any = 0. Since dim zg (x ) = dim zg (h), we have zg (x + y ) zg (x ).
0

(x + y , zg (x + y )) G·(t â t) by induction assumption. Hence (x , q ) G·(t â t) for any q zg (x ).

Corollar y
dim C (g) = dim g + dim t.

Example
son = {skew-symmetric n â n matrices}. Therefore C alt (2, n) is irreducible.
D. Panyushev (Moscow) On the irreducibility of commuting varieties LAW '08 7 / 21


Reductive Lie algebras and commuting varieties

Richardson's theorem

Related problems on C (g)
Algebraic-geometric proper ties
prove that C (g) is a normal variety prove that C (g) has rational singularities construct a resolution of singularities of C (g) compute the degree of C (g) compute the Hilber t polynomial of C[C (g)]

Prove that the natural quadratic equations generate the ideal of C (g) in C[g â g] What is known: the quotient variety C (g)/ G is normal (A . J O S E P H, 1996) / The degree is computed for g = gl(V ) (A . K N U T S O N , P. Z I N N - J U S T I N, 2006), see Sequence A029729. results for small ranks
D. Panyushev (Moscow) On the irreducibility of commuting varieties LAW '08 8 / 21


Generalisations

Triples of matrices

d -tuples of commuting matrices
Definition
C (d , n) = {(A1 , . . . , Ad ) | [Ai , Aj ] = 0 i , j } (gln ) C (d , n) is reducible for n 4, d
d

5 (Gerstenhaber, 1961);

C (d , 2), C (d , 3) are irreducible for any d , C (4, 4) is reducible (Kirillov­Neretin, 1984); C (3, n) is reducible for n C (3, n) is irreducible for n Omladic, Han, Sivic.) 32 (Guralnick, 1992), now n 30; 8 (Guralnick-Sethuraman, Holbrook,

For more details, attend Sivic's talk tomorrow !

D. Panyushev (Moscow)

On the irreducibility of commuting varieties

LAW '08

9 / 21


Generalisations

Non-reductive subalgebras of gln

Triangular matrices­1

b = {upper-triangular n â n matrices} t = {diagonal matrices}. Then B ·t = b. It can be shown that B ·(t â t) is always an irreducible component of C (b).

Problem
Is it true that C (b) = B ·(t â t) ? The general answer is "no".

D. Panyushev (Moscow)

On the irreducibility of commuting varieties

LAW '08

10 / 21


Generalisations

Non-reductive subalgebras of gln

Triangular matrices­2
Example
n = 3m, dim B ·(t â t) = dim + dim b 0 Take a = 0 0 00 t = 9 (m2 + m). 2 b. 0 5m2 .

Then dim a = 3m2 and dim[a, a] = m2 . Therefore dim (a â a) C (b) = dim C (a) Hence C (a) B ·(t â t) for m

10 and C (b) is reducible.

Related problem
Determine parabolic subalgebras p gln having the proper ty that C (p) is irreducible.
D. Panyushev (Moscow) On the irreducibility of commuting varieties LAW '08 11 / 21


Generalisations

Non-reductive subalgebras of gln

Centralisers of nilpotent elements of gln -- I
The nilpotent orbits are parametrised by par titions of n. Let = (1 , . . . , s ) be a par tition of n. (Notation: n.) Then O() and z() denote the corresponding orbit and centraliser, respectively.

Examples
if = (n), then O() is regular and z() is abelian, if = (2, 1, . . . , 1), then O() is the minimal (nonzero) orbit.

Theorem (M.G. Neubauer and B. A. Sethuraman, 1999)
If has two nonzero par ts, then C (z()) is irreducible. If a matrix is 2-regular and nilpotent, then it has at most two Jordan blocks.
D. Panyushev (Moscow) On the irreducibility of commuting varieties LAW '08 12 / 21


Generalisations

Non-reductive subalgebras of gln

Centralisers of nilpotent elements of gln -- II

There is a connection between C (z()) and commuting triples:

Theorem (O. Yakimova, 2006)
If C (z()) is irreducible for any irreducible as well. m with m n, then C (3, n) is

Corollar y
For n 30, there is a n such that C (z()) is reducible.

However, no explicit examples is known.

D. Panyushev (Moscow)

On the irreducibility of commuting varieties

LAW '08

13 / 21


Generalisations

Nilpotent commuting varieties

Nilpotent commuting varieties ­ I
N g ­ the cone of nilpotent elements.

Definition
The nilpotent commuting variety is C (N ) := C (g) (N â N ).

Theorem (A. Premet, 2003)
(i) The irreducible components of C (N ) are parametrised by the distinguished nilpotent G-orbits in g. (ii) The variety C (N ) is of pure dimension dim g. Consider the projection p : C (N ) N If O N is an orbit, then dim p-1 (O) dim g and the equality exactly means that O is distinguished.

D. Panyushev (Moscow)

On the irreducibility of commuting varieties

LAW '08

14 / 21


Generalisations

Nilpotent commuting varieties

Nilpotent commuting varieties ­ II

Corollar y (Baranovsky, 2001; Basili, 2003)
For g = sln , the variety C (N ) is irreducible. (The only other cases are so5 and so7 .) It is an interesting problem to describe nilpotent matrices commuting with a given nilpotent matrix ( talk of P. Oblak tomorrow).

D. Panyushev (Moscow)

On the irreducibility of commuting varieties

LAW '08

15 / 21


Generalisations

Knutson's diagonal scheme

The diagonal commutator scheme
Suppose g is semisimple = [ , ] : g â g g ­ the usual bracket and C (g) = The expected dimension of
-1 -1

(0).

is onto and the generic fibre of is of dimension dim g. (t) is dim g + dim t.

Theorem (Knutson, 2005)
If g = sln , then -1 (t) is a reduced complete intersection. It has two irreducible components of dimension dim g + dim t. Knutson constructs a degeneration of -1 (t) into the scheme {(A, B ) | AB is upper triangular, BA is lower triangular}. The latter has n ! irreducible components, which are parametrised by permutations. Knutson also studies the degree of irreducible components.
D. Panyushev (Moscow) On the irreducibility of commuting varieties LAW '08 16 / 21


Involutions and commuting varieties

Involutions and commuting varieties
For an involution Aut(g), let g = g0 g1 be the corresponding Z2 -grading. Then (g, g0 ) is called a symmetric pair.

Definition
The commuting variety associated with is C (g1 ) = C (g) (g1 â g1 ). C (g1 ) is the zero fibre of 1 : g1 â g1 g0 ; The group G0 acts on g1 and C (g1 ); If c is a Car tan subspace of g1 , then G0 ·c = g1 . G0 ·(c â c) is an irreducible component of C (g1 ). dim G0 ·(c â c) = dim g1 + dim c. The induction scheme of Richardson basically applies here. The problem reduces to study of -distinguished (nilpotent) G0 -orbits in g1 .
D. Panyushev (Moscow) On the irreducibility of commuting varieties LAW '08 17 / 21


Involutions and commuting varieties

Involutions of maximal rank
We say that is of maximal rank if dim c = rk g. In this case c is Car tan and dim g1 - dim g0 = rk g.

Theorem (Panyushev, 1994)
If is of maximal rank, then C (g1 ) is an irreducible normal complete intersection. The ideal of C (g1 ) in C[g1 â g1 ] is generated by quadrics. 1 is onto, hence dim C (g1 ) dim G0 ·(c â c) = dim g1 + rk g; The complement of G0 ·(c â c) forms a subvariety of dimension less than dim g1 + rk g; Here 1 : g1 â g1 g0 is flat and all the fibres are irreducible, normal, etc. 2 dim g1 - dim g0 = dim g1 + rk g;

D. Panyushev (Moscow)

On the irreducibility of commuting varieties

LAW '08

18 / 21


Involutions and commuting varieties

Some examples
Example
For g = gln , the involution of maximal rank is given by (A) = -At . It follows that g0 = son and g1 = {symmetric n â n matrices}. Therefore C
sym

(2, n) is irreducible. C (g1 ) is not always irreducible.

Unpleasant fact:

Example (Panyushev and Yakimova, 2007)
s = diag(1, . . . , 1, -1, . . . , -1) GLn
m n +m

and = Int(s). Then , and dim c = min{n, m}.

g0 =

0 0

glm gln , g1 =

0 0

Here C (g1 ) is reducible unless n = m.
D. Panyushev (Moscow) On the irreducibility of commuting varieties LAW '08 19 / 21


Involutions and commuting varieties

The rank one case
Theorem (Panyushev, 2004)
If dim c = 1, then #Irr(C (g1 )) = #{nonzero G0 -orbits in N g1 }.

Number of irreducible components
(son , son-1 ) n3 1 (sp2n , sp2n-2 â sp2 ) n2 2 (F4 , so9 ) 2 (sln , sln-1 â T1 ) n3 3

Fact: The standard component G0 ·(c â c) is always a unique irreducible component of maximal dimension. If dim c = 1, then all other components are of dimension dim g1 .

D. Panyushev (Moscow)

On the irreducibility of commuting varieties

LAW '08

20 / 21


Some open problems

Open problems

For what par titions is the variety C (z()) irreducible? Study the triples of commuting nilpotent matrices. Study the triples of commuting symmetric matrices. Is the variety {(A1 , A2 , A3 ) C (3, n) | A1 A2 = A2 } irreducible ? 3 Describe Irr C (g1 ) for the symmetric pair (gl
n +m

, gln â glm ), n = m.

The irreducibility of C (g1 ) is not known for 3 cases. Two serial cases concern classical symmetric pairs: (so4n , gl2n ) and (sp2n+2m , sp2n â sp2m ), min(n, m) 3. There is no general principle for the irreducibility of C (g1 ) !

D. Panyushev (Moscow)

On the irreducibility of commuting varieties

LAW '08

21 / 21