By Brian D. Boe

This booklet investigates the composition sequence of generalized vital sequence representations brought on from a maximal cuspidal parabolic subgroup of a true reductive Lie workforce. Boe and Collingwood examine whilst such representations are multiplicity-free (Vogan's challenge #3) and the matter of describing their composition elements in closed shape. the consequences received are strikingly just like these of Enright and Shelton for optimum weight modules. Connections with diversified flag sort decompositions are mentioned.

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**Additional info for Enright-Shelton Theory and Vogan's Problem for Generalized Principal Series**

**Example text**

J - i is odd. \ € ftp. 7), e = par^(e i + 1 - ek) = par*(ej - e,_ 1) = -/x. D The next several results provide specific information about the correspondence between sets ft € £s and subsets ( S * ^ . These results will be crucial for proving that the composition factors of TT(8) occur without multiplicities. We begin by showing that ft+ uniquely determines ft. 10. Given 8 e P™ax and a subset G C $ + , there exists at most one ft G £$ such that ft j = 0 . PROPOSITION Proof. Suppose such an ft exists.

5 p + 1 X % ) ) . 19a) V0 = VG U VPn U P P n _ 1 U VPrn, leading to parameters of types 2* in T>G\ (i,j) and (z,j) in Vpn\ i in Dp n _jj (i,j)* and (2,j) b i n P P m . 19b) S = {1*} U {(0, j) | 1 < j < 2n - 2} is in one-to-one correspondence with the diagram W^ with its longest element removed. )• Lot V$ denote the totality of Qtt-orbit or K-orbit parameters attached to S in this way. ,„-1)1 } 44 BRIAN D. BOE AND DAVID H. COLLINGWOOD We can arrange things so that 6\+ indexes the holomorphic discrete series representation in HC0.

3, consider the set S' consisting of all z e Phoio,M such that (1,2) ^ z :< (n - l,n) . 22b) S = S'u{l}. Then S parametrizes a small part of the Hermitian symmetric poset W^ attached to the pair (D n , An-\). )• Let P 5 denote the totality of Q^-orbit or K-orbit parameters attached to S in this way. 4): wi if 2 < j < n, = Si] 48 BRIAN D. BOE AND DAVID H. COLLINGWOOD if 3 < j < n, 2*
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