Standard Parabolic Subgroup Research Articles

Toda lattice and toric varieties for real split semisimple Lie algebras

The paper concerns the topology of an isospectral real smooth manifold for certain Jacobi element associated with real split semisimple Lie algebra. The manifold is identified as a compact, connected completion of the disconnected Cartan subgroup of the corresponding Lie group G which is a disjoint union of the split Cartan subgroups associated to semisimple portions of Levi factors of all standard parabolic subgroups of G. The manifold is also related to the compactifled level sets of a generalized Toda lattice equation defined on the semisimple Lie algebra, which is diffeomorphic to a toric variety in the flag manifold G/B with Borel subgroup B of G. We then give a cellular decomposition and the associated chain complex of the manifold by introducing colored-signed Dynkin diagrams which parametrize the cells in the decomposition.

On pointwise conjugacy of distinguished coset representatives in Coexeter groups

Let (W, S) be a Coxeter system. For a standard parabolic. subgroup W-K, K subset of or equal to S let D-K be the set of distinguished coset representatives, i.e. representatives of cosets W(K)w of minimal Coxeter length. If L = K-c subset of or equal to S with c is an element of W, then D-K and D-L = c(-1) D-K are in general not conjugate as sets. However it is shown that if WK is finite, they are conjugate 'pointwise', i.e. there is a bijection theta : D-K --> D-L such that theta(d) = d(wc) for some w is an element of W-K depending on d is an element of D-K. In particular for each conjugacy class C of W the cardinalities # (D-K boolean AND C) and # (D-L boolean AND C) are the same. The case of infinite standard parabolic subgroups is also discussed and a corresponding result is proved.

Some remarks on degenerate principal series

In this paper, we give a criterion for the irreducibility of certain induced representations, including, but not limited to, degenerate principal series. More precisely, suppose G is the F -rational points of a split, connected, reductive group over F , with F = R or p-adic. Fix a minimal parabolic subgroup Pmin = AU G, with A a split torus and U unipotent. Suppose M is the Levi factor of a parabolic subgroup P Pmin, and an irreducible representation of M. Further, we assume that has Langlands data (A; ) in the subrepresentation setting of the Langlands classication (so that ,!Ind Mmin\M ( 1)). The criterion gives the irreducibility of Ind G ( 1) if a collection of induced representations, induced up to Levi factors of standard parabolics, are all irreducible. This lowers the rank of the problem; in many cases, to one. The approach used to obtain our criterion is based on an argument in [Tad1]. We note that, since F may be real or p-adic, this gives an instance of Harish-Chandra’s Lefschetz principle in action (cf. [H-C]). We now discuss the contents, section by section. In the rst section, we review notation and give some background. The second section contains the main irreducibility result (cf. Theorem 2.6). This tells us the induced representation is irreducible if three conditions { denoted (), (), ( ){ all hold. As mentioned above, these conditions involve the irreducibility of representations induced up to Levi factors of standard parabolic subgroups. In the third section, we show that () implies () (cf. Proposition 3.3). In the fourth section, we give a more explicit description of () when G = Spn(F) (cf. Lemma 4.1), and use this to show that () implies () for G = Spn(F) (cf. Proposition 4.2). To show that () implies () for G = Spn(F), we use the results of [Gol], so that, at this point, we assume that if F is p-adic, charF = 0. We also discuss other cases where () implies (), and give an example where it does not. In the fth section, we apply these results to the example of degenerate principal series for Spn(F), i.e., P = MU is maximal and is one-dimensional (cf. Corollary 5.2). We also compare the results to known results for degenerate principal series for Spn(F), to

The First Term Identities for Eisenstein Series

In his celebrated paper [We], A. Weil proved the Siegel formula for classical groups, which is called the Siegel Weil formula. The Siegel Weil formula is an identity which relates the value of Siegel Eisenstein series and the average of theta series under certain convergence condition. In order to apply the Siegel Weil formula to the study of special values of automorphic L-functions and automorphic representations, S. Kudla and S. Rallis have extended the Siegel Weil formula beyond the region of convergence, by means of regularization. This extension is expected to work for all classical groups. The regularized Siegel Weil formula by Kudla and Rallis is also called the First Term Identity for Eisenstein series because it is turned out to be an identity relating the first term of the Siegel Eisenstein series and the first term of another Eisenstein series of non-Siegel type, by means of the Laurant expansion of Eisenstein series at certain point. We are interested in characterizing the first terms of Eisenstein series attached to all maximal parabolic subgroups and establishing all possible first term identities for Eisenstein series of the symplectic group. Because our method is not restrictive, it is natural to expect that the similar general first term identities should hold for other groups, at least for classical groups. We shall explain below some more details. Let F be a number field and A=AF the adele ring of F. Let Gn :=Sp(2n) be the symplectic group of rank n, and Pr =M n r N n r be the standard maximal parabolic subgroup of Gn with the Levi part M r isomorphic to GL(r)_Gn&r (m=m(ar , gn&r)). We denote by I r (s) the degenerate principal series representation of Gn(A) induced (normalized induction) from the one dimensional representation |det ar | s of Pr(A). Then, as usual, for any section fs # I r(s), one can form an Eisenstein series as follows:

Convex Orderings in Affine Root Systems, II

This paper contains several results concerning convex orderings in root systems. These results complete the analysis developed in earlier papers by the author. We obtain, in the finite case, a combinatorial characterization of reflections, longest elements in standard parabolic subgroups of the Weyl group of the root system, and of Coxeter elements. In the affine case, a refinement of the main construction done earlier by the author (J. Algebra172, 1995, 613–623) is described; moreover, infinite periodic compatible sets are classified.

Boundaries of Coxeter Matroids

The purpose of this paper is to introduce, for a finite Coxeter groupW, the mod2 boundary operator on the space of all Coxeter matroids (also known asWP-matroids) forWandP, wherePvaries through all the proper standard parabolic subgroups ofW(Theorem 3 of the paper). A remarkably simple interpretation of Coxeter matroids as certain sets of faces of the generalized permutahedron associated with the Coxeter groupW(Theorem 1) yields a natural definition of the boundary of a Coxeter matroid. The latter happens to be a union of Coxeter matroids for maximal standard parabolic subgroupsQiofP(Theorem 2). These results have very natural interpretations in the case of ordinary matroids and flag-matroids (Section 3).

The Noncommutativity of Hecke Algebras Associated to Weyl Groups

We prove that the Hecke algebra $\mathcal {H}(W,{W_J})$, where W is a Weyl group of spherical type and ${W_J}$ is a standard parabolic subgroup of W of corank $\geq 2$, is noncommutative.

On Eisenstein Series of GL n

In this paper, we present a new method of obtaining an analytic continuation and explicit functional equation for an Eisenstein series attached to any standard maximal parabolic subgroup of GL n . Moreover, our method yields some information about the location of possible poles of our Eisenstein series.

The noncommutativity of Hecke algebras associated to Weyl groups

We prove that the Hecke algebra H ( W , W J ) \mathcal {H}(W,{W_J}) , where W is a Weyl group of spherical type and W J {W_J} is a standard parabolic subgroup of W of corank ≥ 2 \geq 2 , is noncommutative.

Residual automorphic representations of Sp4

Let G = Sp4 be the symplectic group of degree two defined over an algebraic number field F and K the standard maximal compact subgroup of the adele group G (A). By the general theory of Eisenstein series ([14]), one knows that the Hilbert space L2(G(F)\G(A)) has an orthogonal decomposition of the formL2(G(F)\G(A)) = L2(G) ⊕ L2(B) ⊕ L2(P1) ⊕ L2(P1),where B is a Borel subgroup and Pi are standard maximal parabolic subgroups in G for i = 1,2. The purpose of this paper is to study the space L2d(B) associated to discrete spectrurns in L2(B).

Twisted endoscopy and reducibility of induced representations for p-adic groups

1. Introduction. This is the first in a series of papers in which we study the reducibility of representations induced from discrete series representations of the Levi factors ofmaximal parabolic subgroups of p-adic groups on the unitary axis. The problem is equivalent to determining certain local Langlands L-functions which are of arithmetic significance by themselves. One of the main aims of this paper is to interpret our results in the direction of the parametrization problem by means of the theory of twisted endoscopy for which our method seems to be very suitable. When the representation is supercuspidal, we also study the reducibility of the induced representations off the unitary axis. Let F be a p-adic field of characteristic zero. Fix a positive integer n > 1 and let G be either of the groups SP2n, S02n or S02n+l. In all three cases there is a conjugacy class of maximal parabolic subgroups which has GL,, as its Levi factor. Let P MN be the standard parabolic subgroup ofG in this conjugacy class. Then M GL,,. Let tr be a discrete series representation of M GLn(F) and, given s C, let I(s, tr) be the representation of G G(F) induced from tr (R) Idet( )ls. Let I(tr) I(0, tr).

Pre-Tits Systems

AbstractA Tits system gives rise to the corresponding pre-Tits system which is the free product of the standard parabolic subgroups of rank 1 amalgamating B. Some properties of this system are investigated.

Local coefficients as Artin factors for real groups

Introduction. The purpose of this paper is to prove the equality of certain local coefficients of arithmetic significance which were attached to representations of quasi-split real reductive algebraic groups in [27] with their corresponding Artin factors attached by local class field theory [21]. As a consequence, we establish an identity satisfied by certain normalized intertwining operators. It seems to be useful in applications of the trace formula [1, 29]. More precisely, let G be the group of real points of a quasi-split reductive algebraic group over R. Let A be the set of simple roots defined by a fixed minimal parabolic subgroup Po = M0A0U of G. Fix 6 A, and let P = P9 be the corresponding standard parabolic subgroup of G and write P = MAN for its Langlands decomposition. Fix a nondegenerate character x °f U i ^ (a, H{o)) be an irreducible admissible x-gic Banach (in particular x~generic unitary) representation of M (cf. Section 1). Given v G a£, the complex dual of the Lie algebra of A, let I(v,o,0) be the continuously (quasi-unitarily, if o is unitary) induced representation IndP^Ga® e v 9 and let V(v,o,0) be its space (Section 0). Then F ^ a , ^ = Vfoo^O). Now, let W be the Weyl group of Ao in G. Choose w G W such that w(0) A. Let N~ be the unipotent group opposite to N. Define N$ = U C wN~w~ where w is a representative of w in G. For/ G F(^,a,^)00, define

Parabolic subgroups of twisted Chevalley groups over a semilocal ring

It is shown that, under minor additional assumptions, the standard parabolic subgroups of a Chevalley group G ρ (Φ, R) of twisted type Φ=Al,l odd, Dl, E6 over a commutative semilocal ring R with involution ρ are in one-to-one correspondence with the ρ-invariant parabolic nets of ideals of R of type Φ, i.e., with the sets , of ideals σ α of R such that: (l) whenever ; (2)ρσ α =σ ρα for all α; (3) σ α =R forα > 0. For Chevalley groups of normal types, analogous results were obtained in Ref. Zh. Mat. 1976, 10A151; 1977, 10A 301; 1978, 6A476.