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Some Intersection Properties of the Fibres of Springer's Resolution

James S. Wolper

1984
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Proceedings of the American Mathematical Society
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Combinatorial results are used to calculate the dimension of the intersection of any two irreducible components of the set in the flag variety fixed by the action of a unipotent element of GL" whose Jordan decomposition has two blocks. This is then related to the "left cells" of Kazhdan and Lusztig, which are used to construct representations of S", the Weyl group of GL". 0. In this note, we study the fibres of Springer's resolution [St] of the singularities of the unipotent variety in G =
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... variety in G = GL"(/c), where k is an algebraically closed field. These fibres are fixed point sets for the action of G on the variety 38 parametrizing the complete flags in a vector space of dimension n. We use 38 u to fixed by a unipotent element u e SL". In general, S8U has several irreducible components. Suppose that the Jordan decomposition of u has block sizes X, > X2 ^ • • • > Xs, where a, + ■ ■ ■ + Xs = n. (We refer to this as the shape X, and, by abuse of language, we say that u has shape X.) Then each component of 38 u has dimension 2Zsi=l(i -1)a(, and there is one component for each standard Young tableau of shape X (see [S] and below). However, one does not know, in general, the codimension of the intersection of two components; only the "one-hook" case has been done [V]. Here, we calculate the codimension when the Jordan form of u has two blocks. This calculation depends on combinatorial techniques exposed in [LS]. The precise result is (2.1). This calculation has two-fold significance. First, it enables one, in this case, to verify a conjecture of Kazhdan and Lusztig [KL, 6.3] concerning the configuration of components of 3 §u; see (4.3). Second, this casts new light on the combinatorial results in [LS] which are used to calculate Kazhdan-Lusztig polynomials, because there is no mention of the geometry of the Grassmannian here. 1.0. In this section, we begin to associate (following [LS]) components of 33 u (for certain u) to words in the letters a and ß (and vice-versa). Let M* denote the set of words made up of a a's and b ß's, e.g., M2 = { ßaa, aßa, aaß). We associate a permutation of {1, 2,..., a + b) to each w e Ma* by writing 1,..., a in order from right to left under the a's, and then a + 1,..., a + b in order from right to left under the ß 's. Thus, aaßaßßa is associated to 4, 3, 7, 2, 6, 5, 1, which can be written (14237X56).

doi:10.2307/2044622
fatcat:yytu5u6jpvb6xombpw7i5cwodi