Ascona Conference Algebraic Groups and Invariant Theory Springer fibers admitting singular components Lucas Fresse, joint work with Anna Melnikov
Definition of Springer fibers Let V = C n, let u End(V ) nilpotent. A complete flag is a maximal chain of vector subspaces: F = (0 = V 0 V 1... V n = V ), dim V i = i ( i). B = {complete flags} is an algebraic projective variety. Define: B u = {F B : u(v i ) V i i}. This is a projective subvariety of B, called Springer fiber. Bu is connected, in general reducible.
B u in geometric representation theory Jordan form of u represented by a Young diagram: (λ 1... λ r ) (sizes of the Jordan blocks of u) Y (u) = (λ i boxes in the i-th row) T.A. Springer (1976): link between Bu and the representation theory of S n : structure of Sn -module on H (B u, Q), such that (M(Y ): Specht module). H max (B u, Q) = M( Y (u) ) Irr(S n )
Particular cases Two simple cases: If Y (u) =, then u is regular and B u = {pt}. If Y (u) = :, then u = 0 and B u = B. In any other case, B u is reducible. It may happen that every component of Bu is nonsingular: in the hook case, i.e. Y (u) = : (J.A. Vargas, 1979) in the 2-row case, i.e. Y (u) = (F. Fung, 2003). However, for Y (u) = Bu has a singular component (J.A. Vargas, 1979).
Main result Theorem. Every irreducible component of B u is nonsingular exactly in four cases: (i) the hook case Y (u) = : (ii) the 2-row case Y (u) = (iii) the 2-row plus one box case: Y (u) = (iv) (exceptional case) Y (u) =.
Preliminaries: B u and the combinatorics of Young Jordan form of u represented by a Young diagram: Y (u) =. (N. Spaltenstein): dim Bu = s j=1 1 2 λ j (λ j 1) where λ 1,..., λ s are the sizes of the columns of Y (u). Standard tableau = numbering of Y (u) by 1,..., n, increasing along the rows and the columns. Example: T = 1 3 6 2 5 4 7 (N. Spaltenstein): the irreducible components of Bu are parameterized by the standard tableaux of shape Y (u)..
B u and the combinatorics of Young (continue) Spaltenstein s construction of the components: T standard gives rise to Y 1 (T )... Y i (T )... Y n (T ) = Y (u), where Y i (T ) = shape of the subtableau T [1,..., i]. F = (V0,..., V n ) B u gives rise to Y (u V1 )... Y (u Vi )... Y (u Vn ) = Y (u). Define Bu T = {F B u : Y (u Vi ) = Y i (T ) ( i)}, we get B u = T BT u. Bu T is locally closed, irreducible, dim Bu T = dim B u ( T ). K T := Bu T are the irreducible components of B u, and dim K T = dim B u ( T ).
1st step: inductive criterion of singularity Let T be standard, K T B u the associated component. Let T = T [1,..., n 1], it gives K T B u component. Theorem. (a) K T is singular K T is singular. (b) Moreover, if n lies in the last column of T, then: K T is singular K T is singular. Skip of the proof. Let U = {F = (V 0,..., V n ) K T : Y (u Vn 1 ) = Y n 1 (T )}. We have B T u U K T, and U is open in K T. In case (b), U is closed, hence U = K T. Thus, it suffices to show: K T singular U singular. To do this, we show that Φ : U H, (V0,..., V n ) V n 1 is a fibre bundle over its image, of base nonsingular, of fiber K T.
2nd step: construction of singular components Two basic singular components: Proposition. (a) If T = (b) If T = 1 2 5 3 4 6 7 1 3 2 5, then K T is singular. (Vargas) 4 6, then K T is singular. Combining with the previous criterion, we get: Corollary. Whenever Y (u) contains or as a subdiagram, B u admits a singular component.
3nd step: the (r, s, 1)-case It remains to show that, in the following cases: (i) hook case Y (u) = : (ii) 2-row case Y (u) = (iii) 2-row plus one box case: Y (u) = (iv) case Y (u) =, every component of B u is nonsingular. In cases (i), (ii), (iv), all the components are nonsingular. Thus, it remains to show: If Y (u) has three rows of sizes r, s, 1, then every component of B u is nonsingular.
3nd step: the (r, s, 1)-case (continue) We may assume r = s. We show that K T (r) is nonsingular for T (r) = 1 2 r r+1 r+2 2r. 2r+1 Say T T if K T, K T are either both singular or both nonsingular. We have T T if T is obtained from T by: adding/deleting n in the last column, the Schützenberger involution T Sch(T ). For T of (r, r, 1)-type, we show T T (r ) for some r r. Therefore K T is nonsingular.
1st step: inductive criterion of singularity Let T be standard, K T B u component. Let T = T [1,..., n 1], it gives K T B u component. Theorem. (a) K T is singular K T is singular. (b) Moreover, if n lies in the last column of T, then: K T is singular K T is singular. In particular, adding one box to a tableau preserves the singularity of the component.