The sheets of a classical lie algebra

We consider the adjoint action of a connected complex semisimple group G on its Lie
algebra g. A sheet of g is a maximal irreducible subset of g consisting of G-orbits of
a fixed dimension. The Lie algebra g is the finite union of its (not necessarily disjoint)
sheets. It is known how sheets are classified, and how they intersect (see [2] for the whole
story).
Let S be a sheet of g. A fundamental result says that S contains a unique nilpotent
orbit. Let {e, h, f} be a standard triple in g such that e is contained in S. Let gf be the
centralizer of f in g and define X � gf by e +X = S \ (e + gf ). Katsylo then constructs
in [9] a geometric quotient : S ! (e+X)/A where A denotes the centralizer of the triple
in G.
On the other hand, Borho and Kraft consider the categorical quotient �
S
: S ! S//G
and the normalization map of S//G. They construct a homeomorphism from the normalization
of S//G to the orbit space S/G, which is equipped with the quotient topology.
Suppose S were smooth (or normal). The restriction of �S
to S then factors through
the normalization of S//G and the induced map is a geometric quotient by a standard
criterion of geometric invariant theory ([15], Proposition 0.2). We note that the induced
map may be a geometric quotient without S being smooth (or normal).
The purpose of this work, however, is to investigate the smoothness of sheets. The
main result is:
Theorem. The sheets of classical Lie algebras are smooth.
If g is sln, this is a result of Kraft and Luna ([13]), and of Peterson ([17]) (see also [1] for
a detailed proof). For the other classical Lie algebras a few partial results were obtained
by Broer ([4]) and Panyushev ([16]). They both heavily use some additional symmetry.
On the other hand, one of the sheets of G2 is not normal (see [19]), the remaining ones
being smooth. For most of the sheets of exceptional Lie algebras it is not known whether
they are smooth or not.
This work is organized as follows:
In the first chapter, we recall the notions of decomposition class and of induced orbit,
as well as their relevance to the theory of sheets. Let l be a Levi subalgebra of g and x 2 l
a nilpotent element. The G-conjugates of elements y = z + x such that the centralizer
of z is equal to l form a decomposition class of g (“similar Jordan decomposition”). The
fact that every sheet contains a dense decomposition class leads to the classification of
sheets by G-conjugacy classes of pairs (l,Ol) consisting of a Levi subalgebra of g and a so
called rigid orbit Ol in the derived algebra of l. A rigid orbit is a (nilpotent) orbit which
itself is a sheet. The unique nilpotent orbit in the sheet corresponding to a pair (l,Ol) as
above is obtained by inducing Ol from l to g: Let p be any parabolic subalgebra of g with
Levi part l, and pu its unipotent radical. The induced orbit Indg
l Ol is then defined as the
unique orbit of maximal dimension in G(Ol + pu).
In the second chapter, we explain Katsylo’s results on sheets in detail. Let S be the
sheet corresponding to a pair (l,Ol) and let {e, h, f} be a standard triple in g such that
e is contained in S. If the triple is suitably chosen the sheet S may be described as
G(e + k) where k denotes the center of l. We use the canonical isomorphism attached to
the triple (2.1), and obtain a morphism ": e + k ! e + gf such that e + z and "(e + z)
are G-conjugate for every z 2 k. It turns out that "(e + k) is an irreducible component of
e + X, the intersection of S and e + gf . Moreover, the centralizer of the triple in G acts
transitively on the set of irreducible components of e + X, and its connected component
acts trivially on e + X. Essentially by sl2 theory, the two varieties S and e + X are
smoothly equivalent. This is the approach we use to investigate smoothness of sheets. At
the end of the chapter, we apply these ideas to the regular sheet of g and to admissible
sheets of g. The regular sheet is the (very well known) open, dense subset consisting of
the regular elements of g. It corresponds to the pair (h, 0) where h is a Cartan subalgebra
of g. By Kostant, e + gf is contained in the regular sheet and every regular element is
G-conjugate to a unique element of e + gf . Hence " maps e + h onto e + gf ; it is the
quotient by the Weyl group of G. The admissible sheets, in this context, are those coming
nearest to the regular sheet.
In the remaining chapters, we deal with sheets in classical Lie algebras (in fact, our
setting is slightly more general (3.1)). We prove that " maps e+k onto e+X; it turns out
to be the quotient by some reflection group acting on k. Therefore e+X is isomorphic to
affine space and so S is smooth.
We first take a look at the linear group, that is, G is equal to GL(V ) for some complex
vector space V . In this case, the sheets of g are in one-to-one correspondence to the
partitions of dim V (3.3). In order to make this explicit, we associate a partition to every
y 2 g as follows: We decompose V as a C[y]-module into a direct sum of cyclic submodules
by successively cutting off cyclic submodules of maximal dimension. The dimensions of
these direct summands define a partition of dim V . The sheets of g are then the sets S(l)
consisting of elements y 2 g with fixed partition l. The crucial observation is the fact
that there is a decomposition of V into direct summands Vi which respects the setting of
the second chapter in the following sense (Chapter 5): Let S be a sheet of g described as
G(e + k) and let ": e + k ! e + gf be the corresponding map. For every y 2 e + k, the
C[y]-module V decomposes into a direct sum of the same cyclic submodules Vi. We find
elements ei and subspaces ki of gi = gl(Vi) such that Gi(ei +ki) is the regular sheet of gi,
and such that e =
P
i ei and k � �iki. Let "i : ei + ki ! ei + gfi
i be the corresponding
maps. Then " is the restriction of
P
i "i to k. But we already know that "i is the quotient
by the Weyl group of Gi. Finally, a straightforward calculation using basic invariants
(power sums) shows that " is the quotient by the normalizer of k in the Weyl group of G,
which in this case acts as reflection group on k. Since the centralizer of the triple {e, h, f} in G is connected, the image of " is equal to e + X.
The proof for the symplectic groups Sp(V ) and for the orthogonal groups O(V ) follows
along the same lines. We begin with a classification of sheets in combinatorial terms (3.4).
Then we use the combinatorial data to decompose V into a direct sum of subspaces Vi
such that a proceeding similar to the linear case is possible (6.1). To be more precise,
V decomposes as C[y]-module into the direct sum of submodules Vi for every y 2 e + k.
These submodules may not be cyclic; however, they decompose into at most two cyclic
submodules. The next step consists of identifying the maps "i : ei + ki ! ei + gfi
i as
quotients by some reflection group acting on ki. The case of Vi decomposing into two
cyclic submodules of different dimension is the core of this work (6.3). It requires a lot
of ad hoc calculation. The two other cases are readily reduced to the case of the regular
sheet (6.2). At last, a calculation using basic invariants shows that " is the quotient by
some reflection group acting on k (6.4).
Acknowledgments. I am grateful to Hanspeter Kraft for arousing my interest in
this subject, for all his valuable suggestions and support during the course of this work,
and for making it possible to stay at the University of Michigan for a year. I got financial
support from the Max Geldner Stiftung, Basel, during that year abroad. Many thanks go
to Pavel Katsylo and Bram Broer for sharing their ideas, to Stephan Mohrdieck
for his constant interest, and to Jan Draisma for numerous helpful conversations.