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Let (G, V) be a representation with either G a torus or (G, V) a locally free stable $\theta$-representation. We study the fiber at 0 of the associated moment map, which is a commuting variety in the latter case. We characterize the cases where this fiber is normal. The quotient (i.e. the symplectic reduction) turns out to be a specific orbifold when the representation is polar. In the torus case, this confirms a conjecture stated by C. Lehn, M. Lehn, R. Terpereau and the author in a former article. In the $\theta$-case, the conjecture was already known but the present approach yield another proof., Comment: 21 pages, comments welcome

An elementary symplectic group w.r.t. an invertible alternating matrix is defined. It is shown that the group of symplectic transvections of a symplectic module coincides with this elementary symplectic group in the free case. Equality of orbit spaces of a unimodular element under the action of the linear group, symplectic group, and symplectic group w.r.t. an invertible alternating matrix is established.

We develop a structure theory for nilpotent symplectic alternating algebras. We then give a classification of all nilpotent symplectic alternating algebras of dimension up to 10 over any field F.

In this paper, we deal with the problem of classifying compact complex solvmanifolds with holomorphic symplectic structures and obtain some structure results which make the classification possible. In particular, we reduce the classification to the nilpotent case with the same dimension, which we call the nilpotent reduction. The same method also works for the real compact solvmanifolds with real symplectic structures. The real six-dimensional case was treated completely. This is one of the major steps to obtain further examples of compact holomorphic symplectic manifolds. For example, the Kodaira–Thurston surface is NOT a complex homogeneous manifold with a transitive Lie group action which keeps the complex structure invariant, but a real solvmanifold with a complex structure.

Let G be a real form of a complex reductive group. Suppose that we are given involutions \sigma and \theta of G. Let H=G^\sigma denote the fixed group of \sigma and let K=G^\theta denote the fixed group of \theta. We are interested in calculating the double coset space H\backslash G/K. We use moment map and invariant theoretic techniques to calculate the double cosets, especially the ones that are closed. One salient point of our results is a stratification of a quotient of a compact torus over which the closed double cosets fiber as a collection of trivial bundles., Comment: 18 pages, typos corrected

Let G be the symplectic group S p 2 n ( k ) ( k an algebraically closed field of characteristic ≠2). It acts in V = k 2 n . The exotic nilcone of G is the nilcone, in the sense of invariant theory, of the G -module V × ∧ 2 V . The paper discusses elementary properties of the G -action on the exotic nilcone.

In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an optimal way. This is called an {\sl adapted} homotopy basis and there is a corresponding {\sl adapted presentation}. We also give a necessary and sufficient condition for a prime order symplectic matrix to be the image of a prime order element in the mapping-class group., Comment: 24 pages; change in title; typos corrected; some theorems revised; computational complexity added; final version of paper to appear in Journal of Algebra

Let G be a connected reductive group. In this paper we are studying the invariant theory of symplectic G-modules. Our main result is that the invariant moment map is equidimensional. We deduce that the categorical quotient is a fibration over an affine space with rational generic fibers. Of particular interest are those modules for which the generic orbit is coisotropic. We prove that they are cofree. This result has been used in another paper (math.SG/0505268) to classify all these modules. Our main tool is a symplectic version of the local structure theorem., Comment: v1: 24 pages; v2: 31 pages, expanded exposition, new introduction, some facts (esp. Thm. 7.2+Corollaries, Thm. 8.4) which were only implicit in v1 are now spelled out

A quantized symplectic oscillator algebra of rank 1 is a PBW deformation of the smash product of the quantum plane with U_q(sl_2). We study its representation theory, and in particular, its category O., 35 pages. Final version, to appear in Journal of Algebra

Let k be a field and n ⩾ 1 an integer. We study the action of the symplectic group over k on the set of alternating forms on k 2 n . We show that the action on pairs of forms can be interpreted in terms of the conjugation action on self-adjoint operators, and obtain some old and new results using this interpretation. In particular, for each k and n, we determine the smallest base size for the action of PGL 2 n ( k ) on the set of non-degenerate skew-symmetric matrices over the field k, modulo scalars.

We study the flatness for the moment map associated to the cotangent bundle of the space of representations of a quiver Q. If the associated moment map of a root is flat, then we call the root a flat root. We first study the flat roots in the fundamental set of a quiver Q. Then we give an explicit description of all the flat roots of Q. This description is obtained by using the natural class of ( − 1 ) -reflections, which are introduced in this paper. We also show that there are only a finite number of flat roots in each orbit under the Weyl-group of the quiver Q.

Denote by e(π) the classical Frobenius-Schur indicator of the irreducible representation (π, V ) with character χ. That is, e(π) = 1 |G| ∑ g∈G χ(g 2), and Frobenius and Schur proved that e(π) = 1 if (π, V ) is a real representation, e(π) = −1 if χ is real-valued, but π is not a real representation, and e(π) = 0 otherwise. Then the conclusion of Theorem 1.1 is equivalent to the statement that every irreducible representation π of G+ satisfies e(π) = 1. Gow obtained the following intriguing result from Theorem 1.1.

Let a connected reductive group G act on the smooth connected variety X. The cotangent bundle of X is a Hamiltonian G-variety. We show that its "total moment map" has connected fibers. This is an expanded version of section 6 of my paper dg-ga/9712010 on Weyl groups of Hamiltonian manifolds., 15 pages

Given a quadratic extension L/K of fields and a regular alternating space (V, f) of finite dimension over L, we classify K-subspaces of V which do not split into the orthogonal sum of two proper K-subspaces. This allows one to determine the orbits of the group SpL(V, f) in the set of K-subspaces of V.

It is known that the symplectic group Sp 2 n ( p ) has two (complex conjugate) irreducible representations of degree ( p n + 1)/2 realized over Q ( −p ) , provided that p ≡ 3 mod 4. In the paper we give an explicit construction of an odd unimodular Sp 2 n ( p ) · 2-invariant lattice Δ( p , n ) in dimension p n + 1 for any p n ≡ 3 mod 4. Such a lattice has been constructed by R. Bacher and B. B. Venkov in the case p n = 27. A second main result says that these lattices are essentially unique. We show that for n ≥ 3 the minimum of Δ( p , n ) is at least ( p + 1)/2 and at most p ( n − 1)/2 . The interrelation between these lattices, symplectic spreads of F p 2 n , and self-dual codes over F p is also investigated. In particular, using new results of U. Dempwolff and L. Bader, W. M. Kantor, and G. Lunardon, we come to three extremal self-dual ternary codes of length 28.

Let Sp n ( Z ) be the symplectic group on a hyperbolic module V of even rank n over the rational integers Z . If n >2 then any element of Sp n ( Z ) is a product of finite number of involutions on V , and Sp n ( Z ) is generated by two elements of small order, one of which is an involution.

Let R denote a commutative local ring with maximal ideal m and residue field K = R m . Let V be a symplectic space over R . In this paper we determine the group automorphisms of the symplectic group Sp n ( V ) when n ⩾ 6, the characteristic of k is not 2, and k is not the finite field of three elements.

Let V be a finite-dimensional vector space, Q a quadratic form and fa = f the bilinear form associated with Q. We assume (V,&) is regular. Then the orthogonal group O(V) is bireflectional, i.e., every isometry in O(V) is a product of two involutory isometries in O(V). This has been shown in [8] if the field of scalars K has characteristic distinct from 2 and in [4] and [5] if char K = 2. The latter papers also establish the bireflectionality for the symplectic group Sp( V), again under the assumption that (V,f) is regular and char K = 2. For char K # 2 the symplectic group is not bireflectional (see [31). We shall extend the results just mentioned. We shall drop the assumptions that V is finite dimensional and that (V,f) is regular, i.e., the vector space V may be infinite dimensional and the radical of V may be distinct from zero. We shall use the notation and the concepts in [2]. For every 7c E Hom(V, V) we define F(n)= {UE V;?m=u} and B(n)= (vz u; ZJ E V}. The spaces F(r) and B(n) are called fix and path of z, respectively. The groups O*(V) = {n E O(V); rad VC F(z) and dim B(n) < co} and Sp *( V) = { 71 E Sp( V); rad V c F(n) and dim B(n) < co ) are called the weak orthogonal and the weak symplectic group, respectively.

According to a theorem of Dickson and Dieudonne (see [l]), a symplectic group over a field # F2 or F3 , has no normal subgroups other than its center (&I}. Several attempts to generalize this theorem over local rings have been quite successful. First, Klingenberg [3] showed that every normal subgroup of a symplectic group over a local ring is a congruence subgroup (with suitable residue class field). H e assumed that the underlying alternating form has unit discriminant. Riehm [5] generalized these latter results by dropping the requirement that the discriminant be a unit. He introduced the concept of tableau which is a matrix of ideals. By using the tableau, he was able to prove that every normal subgroup of the symplectic group is a congruence subgroup in the sense of tableau. The above integral results were obtained under the assumption that the residue class field has characteristic f2 (i.e., 2 is a unit). The purpose of this paper is to generalize the above results to the case when the residue class field has characteristic 2 but not F, (include the case 2 = 0). The only assumption we will make is that the rank of each modular Jordan component of the alternating form 26. This last assumption is largely technical and is made for the purpose of simplifying some calculations. We show that the symplectic group over the integers of a dyadic field does not have the congruence subgroup property. But we are able to characterize all the normal subgroup of S,(L) by using the tableaus (See Theorem 8.4). In addition to the main result, we also obtain some theorems concerning the generators of the congruence subgroups which are listed as 6.3, 6.4, 6.5, and 8.3. The explicit statement of these theorems are omitted here.

Dickson and Dieudonne have shown in [ 1, 51 that the projective symplectic groups over a field contains more than three elements is simple. Attempts at integral analogs of this theorem have been quite successful. In 1963, Klingenberg [6] showed that the symplectic group of a unimodular alternating module over a nondyadic local domain has the congruence subgroup property, i.e., every normal subgroup is a congruence subgroup in the usual sense mod the ideals of the underlying domain. Klingenberg’s work has been further extended in [3, 4, 71 by re axing 1 the restrictions on the alternating forms and on the underlying domain. In 1967, Bass, Milnor, and Serre [2] effected a significant generalization to the integers of a global field. By using high power tools in class field theory, they were able to show that the symplectic group of a unimodular module with rank 24 has the congruence subgroup property. In this paper, we are going to generalize the above result to a nondyadic principal ideal domain and by dropping the assumption of unimodularity we will provide a more constructive method to classify all the normal subgroups of a symplectic group. We are able to show that a subgroup of a symplectic group is a normal subgroup if and only if it is a congruence subgroup in the sense of tableau which is a matrix of ideals of the underlying domain, i.e., the symplectic group has general congruence subgroup property. In addition to the main theorem we also include generation theorems of congruence subgroups and relations among congruence subgroups, and we also determine the structure of the factor groups of two congruence subgroups of the same order.

In R. E. Solazzi, J. Algebra 21 (1972), 91–102 , the automorphisms of the symplectic congruence groups and their corresponding projective groups were found over any integral domain, in dimensions greater than four. The purpose of this paper is to determine these automorphisms in dimension four. In characteristic not 2 the automorphisms of the projective congruence groups have the same characterization as in the previous paper; they are determined by transformation by a symplectic semisimilitude. This is Theorem 2.5. The remaining sections of this paper deal with exceptional automorphisms of the congruence groups that do not preserve transvections and which can occur with four-dimensional congruence groups in characteristic 2.

The classical branching rule for the symplectic group describes the decomposition of the Schur module Lλ of GL(2n, C ), where λ1 ≤ n, into a direct sum of irreducible representations of Sp(2n, C ) under the embedding Sp(2n, C ) ↪ GL(2n, C ). The purpose of this paper is to extend the classical branching rule to the characteristic-free case by means of explicit universal filtrations. As an application of our methods, we describe universal resolutions of the fundamental representations of the symplectic groups in terms of the fundamental representations of the corresponding general linear groups.

We establish some relations between quadratic symplectic Lie superalgebras and Manin superalgebras. Next, we introduce some concepts of double extensions of quadratic symplectic Lie superalgebras and of Manin superalgebras in order to give inductive descriptions of quadratic symplectic Lie superalgebras.

Problem: Given a reductive algebraic group G, find all k-tuples of parabolic subgroups (P_1,...,P_k) such that the product of flag varieties G/P_1 x ... x G/P_k has finitely many orbits under the diagonal action of G. In this case we call G/P_1 x ... x G/P_k a multiple flag variety of finite type. (If P_1 is a Borel subgroup, the partial product G/P_2 x ... x G/P_k is a spherical variety.) In this paper we solve this problem in the case of the symplectic group G = Sp(2n). We also give a complete enumeration of the orbits, and explicit representatives for them. (It is well known that for k=2 the orbits are essentially Schubert varieties.) Our main tool is the algebraic theory of quiver representations. Rather unexpectedly, it turns out that we can use the same techniques in the present case as we did for G = GL(n) in math.AG/9805067.

Let q be a power of an odd prime p. Let Sp(2n, q) denote the symplectic group of degree 2n over GF(q). Let O+(n, q) denote the split orthogonal group of degree n over GF(q), corresponding to a symmetric form of maximal Witt index, and 0 (n, q) denote the non-split orthogonal group. Unless we need to be more specific, we will simply write O(n, q) for either of these two orthogonal groups. In addition, we will often abbreviate these symbols further to O(n) and Sp(2n) whenever q is understood to be fixed. It is known from conjugacy class considerations that all complex irreducible characters of the finite orthogonal groups O(n) are real-valued. The same is true of the symplectic groups Sp(2n, q) provided that q z 1 (mod 4). When q= 3 (mod 4), not all characters of Sp(2n, q) are realvalued. However, as the total number of irreducible characters is a manic polynomial in q of degree n and this is true of the number of real-valued irreducible characters, we can assert that the majority of characters is realvalued in this case. It is of interest to know whether or not the real-valued characters of these families of groups are the characters of representations that are defined over the real numbers. This paper gives a solution to this problem by proving the following theorem.

The extended quadratic residue codes are known to be invariant under a monomial action by the projective special linear group, an action whose permutation part is the ordinary action on the projective line [3, Theorem 3.11 (see also [7], [15], [16]). Th e re p resentations of the special linear group that arise from the codes are among those constructed in [20] (the group coinciding with the symplectic group in the two-dimensional case). It was thought that the algebraic framework used in [20] to produce these representations could also serve as a starting-point for the codes, thus giving the codes and the group action at the same time. The purpose of the present paper is to substantiate that thought. Here is an outline of the paper and its main results: let V be a vector space of even dimension 2n over the finite field GF(p), 4 a power of an odd prime, and let V be endowed with a non-degenerate symplectic (skewsymmetric) form. Let Q(V) be the corresponding symplectic group on V. An algebra A was constructed in [20] that is basically a twisted group algebra of V over a suitable ring, and from A certain representations of Sp( V) were obtained. Section 1 summarizes these results. In Section 2 several functions arising in that development are calculated more explicitly. Their values involve the constant p of 2.1: p2 = 6q, with Q E 6 (mod 4), 6 = 51. To each maximal isotropic subspace of V corresponds a distinct idempotent of A, and the set I of these idempotents is central to the formation of the codes. The multiplication of these idempotents is also considered in Section 2, the most frequently used result being the Lemma 2.3. (2.6 deals with the characters of the representations.) If the symplectic form is scaled by a non-square of GF@), one obtains another algebra A’, and its relation to A is the subject of Section 3. Section 4 contains the construction of the codes and a monomial action

Motivated by the incidence problems between points and flats of a symplectic polar space, we study a large class of submodules of the space of functions on the standard module of a finite symplectic group of odd characteristic. Our structure results on this class of submodules allow us to determine the $p$-ranks of the incidence matrices between points and flats of the symplectic polar space. In particular, we give an explicit formula for the $p$-rank of the generalized quadrangle ${\rm W}(3,q)$, where $q$ is an odd prime power. Combined with the earlier results of Sastry and Sin on the 2-rank of ${\rm W}(3,2^t)$, it completes the determination of the $p$-ranks of ${\rm W}(3,q)$., 22 pages

The purpose of this paper is to describe a module of the symplectic group which is the analogue of the Carter-Lusztig module of the general linear group ( Math. Z. 136 (1974), 193–242). To accomplish this, we use contraction mappings on tensor space which were first introduced by Weyl (“The Classical Groups,” Princeton Univ. Press, Princeton, N. J., 1946).