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We study irreducible restrictions from modules over symmetric groups to subgroups. We get reduction results which substantially restrict the classes of subgroups and modules for which this is possible. Such results are known when the characteristic of the ground field is greater than $3$, but the small characteristics cases require a substantially more delicate analysis and new ideas. This work fits into the Aschbacher-Scott program on maximal subgroups of finite classical groups.

Let $G \subset GL(V)$ be a reductive algebraic subgroup acting on the symplectic vector space $W=(V \oplus V^*)^{\oplus m}$, and let $\mu:\ W \rightarrow Lie(G)^*$ be the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct a canonical desingularization of the symplectic reduction $\mu^{-1}(0)/\!/G$ for classes of examples where $G=GL(V)$, $O(V)$, or $Sp(V)$. For these classes of examples, $\mu^{-1}(0)/\!/G$ is isomorphic to the closure of a nilpotent orbit in a simple Lie algebra, and we compare the Hilbert-Chow morphism with the (well-known) symplectic desingularizations of $\mu^{-1}(0)/\!/G$., Comment: 21 pages, final version, to appear in Math. Z

We prove that for every reductive group G with a maximal torus T and the Weyl group W there is a natural normalization map chi from T^N/W to an irreducible component of the G-character variety of Z^N. We prove that chi is an isomorphism for all classical groups. Additionally, we prove that even though there are no irreducible representations in the above mentioned irreducible component of the character variety for non-abelian G, the tangent spaces to it coincide with H^1(Z^N, Ad rho). Consequently, this irreducible component has the "Goldman" symplectic form for N=2, for which the combinatorial formulas for Goldman bracket hold., to appear in Math. Z., 18 pages, 1 figure

We show that inside the real quaternionic symplectic group \({\mathrm {Sp}}(p,q)\), every element is conjugate to its inverse. Similar result is proved for the real quaternionic orthogonal group \(\mathrm{O }^{*}(2n)\). We also show that no similar result is possible in the p-adic case.

The aim of this paper is to fill a small, but fundamental, gap in the theory of p-adic analytic groups. We illustrate by example that the now standard notion of a uniformly powerful pro-p group is more restrictive than Lazard’s concept of a saturable pro-p group. For instance, the Sylow-pro-p subgroups of many classical groups are saturable, but need not be uniformly powerful. Extending work of Ilani, we obtain a correspondence between subgroups and Lie sublattices of saturable pro-p groups. This leads to applications, for instance, in the subject of subgroup growth.

If the characteristic of F is zero, this problem has been solved by Saxl [16]. An important feature of Saxl’s result is that a groupG as in the problem above is either 2-transitive or fixes a point, i.e. is contained in some Σn−1. The caseG ≤ Σn−1 can then be settled using the branching rule and induction. On the other hand, an explicit list of k-transitive groups (for k ≥ 2) is available, which can be used to complete the proof in characteristic zero, see [16] for more details. From now on we assume that p > 0. In this case the problem is important for determining maximal subgroups of finite classical groups [1],[13]. However, the situation is nowmore complicated. For example, to determine the pairs (G,D) as above withG = An one needs the Mullineux conjecture [7], [2]. IfG is intransitive then, up to a conjugation, it is contained in a standard Young subgroup of the form Σn−k ×Σk. The irreducible restrictions from Σn toΣn−1 have been described in [14], see also [11], [6]. In [11] it is also

Let G be a complex Lie group and H ~ G be a closed complex subgroup of G. We refer to the quotient X = G/H as a complex-homogeneous manifold. It is natural to ask under which conditions a complex homogeneous manifold X is Stein. First we give some known results. Matsushima [M] proved that if G is reductive, then X = G/H is a Stein manifold, if and only if the group H is reductive. In particular if dime H = 0, then G/H is Stein if and only if H is a finite group. Barth and Otte [BO] showed that for G reductive a complex homogeneous manifold G/H being holomorphically separable implies that H is algebraic in G. As the following example shows, the condition of holomorphical separability is not sufficient for Steinness: Let G = SL2(C)

Using a cohomological obstruction, we construct examples of absolutely simple adjoint classical groups of type 2A_n with n equivalent to 3 mod 4, C_n or 1D_n with n equivalent to 0 mod 4, which are not R-trivial hence not stably rational.

Let V be a form space. If W is a subspace of V we put W {ve Vl~(v, W) =0}. The form space V is called non-defective if V==0. It is called nondegenerate if dim(V=)< 1 and ~(v)#:0 for all non-zero wV  1.2. Assume that V is non-degenerate. V is defective if and only if char(k)=2 and e=0 and dim(V) is odd. If e= 1 then dim(V) is even. Let G = G(V) be the algebraic group in the sense of [-1] of the automorphisms geGl(V) which leave the forms fi and c~ invariant. Let g = g(V) be its Lie algebra. If e = 1 then G is the symplectic group Sp(V), so it is connected and semi-simple. If e=0 then G is the orthogonal group O(V), which is reductive. O(V) has two components if V is non-defective. O(V) is connected if V is defective. 1.3. Our object is to describe the conjugacy classes of the unipotent elements of G and of the nilpotent elements of g under the adjoint action of G given by

In Problem XVII from [4], Weisfeiler indicates a general formulation for homomorphisms between classical groups in terms of a congruence subgroup coming from an induced integral structure in the group. As he observes, certainly not all homomorphisms arise in this manner, but this formulation provides a good foundation for extending the fairly well understood situation for isomorphisms. We establish here Weisfeiler's conjecture for unitary groups, and homomorphisms which do not kill involutions. For monomorphisms this problem is usually solved via group theoretical arguments using derived groups and centralizers, but for homomorphisms these methods are hard to apply and it seems necessary to use a more geometrical approach. In the proof here the central new idea involves moving subspaces in a controlled manner, intersecting them and counting dimensions. It should also be possible to use this technique in other situations. Let V and W be right vector spaces over the division rings k and K with involutions (both denoted by *) where

So we only consider polarizations of nilpotent elements. A more general definition leads to questions which are usually first reduced to the nilpotent case, cf. the proof of [3] (5.16). The structure of Pol(x) is interesting for the infinite-dimensional representation theory, cf. [7] and [1] (6.3).

Let r > 0 be a prime integer, and let G be a finite general linear group GL(n, q), a unitary group U(n, q) with q = qg, or a symmetric group S(n) of degree n. The partition of the irreducible (ordinary) characters of S(n) into r-blocks of S(n) was given by Brauer's and Robinson's solution of Nakayama's conjecture, see [7], p. 245. In their fundamental paper [5] Fong and Srinivasan have recently classified the r-blocks of GL(n, q) and U(n, q) for all primes r > 2 with (r, q)= 1. Using these classifications of the r-blocks of G we show in this article that there is a natural one-to-one correspondence ~b between the irreducible characters of height zero of an r-block B of G with defect group R and the irreducible characters of height zero of the Brauer correspondent b of B in N=NG(R ) (Theorem (4.10)). In particular, ko(B)=ko(b), where ko(B ) denotes the number of all irreducible characters )~ of B with height h t z = 0 . Therefore Alperin's conjecture on the numbers of irreducible characters of height zero is verified for all general linear, unitary and symmetric groups. If G=S(n), then Theorem(4.10) also holds for r = 2. In order to establish the character correspondence ~, three other correspondences are studied, the product of which is ~b. In Sect. 1 we construct for every r-block B of G with defect group R a subgroup G of G with a Sylow r-subgroup /~-~R such that there is a natural height preserving one-to-one correspondence T between the set Irr(B) of all irreducible characters of B and the set Irr(/~0) of all irreducible characters of the principal r-block/~o of d (Reduction Theorems (1.9) and (1.10)). The map ~ respects the geometric conjugacy classes of characters. If /~0 denotes the principal r-block of N=Nd(/~), and if b is the Brauer correspondent of B in N=N6(R), then by Theorems (3.8) and (3.10) the block ideals/~o and b are Morita equivalent, have the same decomposition numbers, and there is a natural height preserving one-to-one correspondence a between Irr(bo) and Irr(b).