A Note on Bruhat Decomposition ofGL(n) over Local Principal Ideal Rings

Abstract

Let A be a local commutative principal ideal ring. We study the double coset space of GLn(A) with respect to the subgroup of upper triangular matrices. Geometrically, these cosets describe the relative position of two full flags of free primitive submodules of A n . We introduce some invariants of the double cosets. If k is the length of the ring, we determine for which of the pairs (n, k) the double coset space depends on the ring in question. For n = 3, we give a complete parametrisation of the double coset space and provide estimates on the rate of growth of the number of double cosets.1.1. Related problems. Let P 1 , P 2 < G be finite groups. Let ρ i = Ind G P i 1 = C[G/P i ] be the representation of G induced from the trivial representation of P i over C. The module of intertwining operators Hom G (ρ 1 , ρ 2 ) can be identified with the subalgebra C[P 2 \G/P 1 ] of left-(P 2 , P 1 )-invariant elements in the group algebra C[G] via the map C[P 2 \G/P 1 ] → Hom(ρ 1 , ρ 2 ) given by f → T f for each f ∈ C[P 2 \G/P 1 ], 2000 Mathematics Subject Classification. 15A33,15A21.

Branching Rules for Principal Series Representations of SL(2) over a p-adic Field

“…Recall that for any u ∈ k × , if X = X(u, 0) then T (X) = ZU for U the unipotent upper triangular subgroup of K. whence (28). For the final statement, we use the explicit form (26) to compute…”

“…Recently, U. Onn and P. Singla completed the study of branching rules of unramified principal series of GL 3 (k) by giving an explicit description of the decomposition into irreducible representations of K [27]. The work with GL 3 (k), along with related calculations by U. Onn, A. Prasad and L. Vaserstein on sizes of double coset spaces in [26], makes it clear that the question of decomposing principal series in the general case will be highly nontrivial.…”

Branching rules for unramified principal series representations of GL(3) over a p-adic field

“…These, and indeed the double cosets in the case where c = (c, c, c) = d, have been described by Onn, Prasad and Vaserstein [9]. Let W = {1, s 1 , s 2 , s 1 s 2 , s 2 s 1 , w 0 } denote the group of permutation matrices in K where s i corresponds to the transposition (i i + 1) and w 0 is the element of maximal length.…”

Geometric interpretation of Murphy bases and an application

“…We remark that for the groups GL n (o 2 ), it is known that the dimensions of complex irreducible representations and their numbers in each dimension depend only on the cardinality of residue field of o, see [11]. For the current setting we shall prove that the numbers and multiplicities of the irreducible constituents of F λ with λ ⊂ L (1) are independent of the residue field as well, though this is not true in general, see [10,3]. In this section we shall use the notation G to denote the group GL n (o 2 ), and the group of invertible matrices of order n over the field k is denoted by GL n (k).…”