On existence of finite universal Korovkin sets in the centre of group algebras

In this paper, we characterize compact groupsG as well as connected central topological groupsG for which the centreZ(L ¹(G)) admits a finite universal Korovkin set. Also we prove that ifG is a non-connected central topological group which has a compact open normal subgroupK such thatG=KZ, thenZ(L ¹(G)) admits a finite universal Korovkin set if is a finite-dimensional separable metric space or equivalentlyG is separable metrizable andG/K has finite torsion-free rank.     

Langlands classification for real Lie groups with reductive Lie algebra

LetG be a (not necessarily connected) real Lie group with reductive Lie algebra. We consider representations ofG which some call admissible but we call them of Harish-Chandra type. We show that any nontempered irreducible Harish-Chandra type representation ofG is infinitesimally equivalent to the Langlands quotient obtained from an essentially unique triple (M, V, ) of Langlands data; while for tempered irreducible Harish-Chandra type representations we prove they are infinitesimally subrepresentations of some induced representations U^V, with imaginary and withV from the quasi-discrete series of a suitableM (perhapsG=M; we define the quasi-discrete series in Definition 4.5 of this paper.We show that irreducible continuous unitary representations of really reductive groups are of Harish-Chandra type. Then the results above yield the canonical decomposition of the unitary spectrum>G for any really reductiveG. In particular, this holds ifG/G ₀ is finite, so the center of the connected semi-simple subgroup with Lie algebra [g, g] may be infinite!Research supported, in part, by the Hungarian National Fund for Scientific Research (grant Nos. 1900 and 2648).     

On Algebras of Invariants and Codimension 1 Luna Strata for Nonconnected Groups

In order to describe explicitly the algebra of invariants for a non-connected reductive subgroup G GL(V) we apply the method of strata. For this we describe codimension 1 strata of the quotient V//G and study the normality property of their closures. We find some criteria for k[V]^G to be polynomial or a hypersurface. Then we apply these results to complete the classification [Sh] of nonconnected simple groups G such that k[V]^G is polynomial.     

The Langlands classification for non-connected -adic groups II: Multiplicity one

For a non-connected reductive -adic group, we prove that the Langlands subrepresentation appears with multiplicity one in the representation parabolically induced from the corresponding Langlands data.

     

Tessellations of Homogeneous Spaces of Classical Groups of Real Rank Two

Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a tessellation if there is a discrete subgroup of G, such that acts properly discontinuously on G/H, and the double-coset space \G/H is compact. Note that if either H or G/H is compact, then G/H has a tessellation; these are the obvious examples.It is not difficult to see that if G has real rank one, then only the obvious homogeneous spaces have tessellations. Thus, the first interesting case is when G has real rank two. In particular, Kulkarni and Kobayashi constructed examples that are not obvious when G=SO(2, 2n)° or SU(2, 2n). Oh and Witte constructed additional examples in both of these cases, and obtained a complete classification when G=SO(2, 2n)°. We simplify the work of Oh-Witte, and extend it to obtain a complete classification when G=SU(2, 2n). This includes the construction of another family of examples.The main results are obtained from methods of Benoist and Kobayashi: we fix a Cartan decomposition G=K A ⁺ K, and study the intersection (K H K)A ⁺. Our exposition generally assumes only the standard theory of connected Lie groups, although basic properties of real algebraic groups are sometimes also employed; the specialized techniques that we use are developed from a fairly elementary level.     

A geometric approach to complete reducibility

Let G be a connected reductive linear algebraic group. We use geometric methods to investigate G-completely reducible subgroups of G, giving new criteria for G-complete reducibility. We show that a subgroup of G is G-completely reducible if and only if it is strongly reductive in G; this allows us to use ideas of R.W. Richardson and Hilbert–Mumford–Kempf from geometric invariant theory. We deduce that a normal subgroup of a G-completely reducible subgroup of G is again G-completely reducible, thereby providing an affirmative answer to a question posed by J.-P. Serre, and conversely we prove that the normalizer of a G-completely reducible subgroup of G is again G-completely reducible. Some rationality questions and applications to the spherical building of G are considered. Many of our results extend to the case of non-connected G. Mathematics Subject Classification (2000) 20G15, 14L24, 20E42     

Triangular actions onC 3

Free algebraic actions of a connected algebraic groupG onC ³ which can be triangularized are shown to be trivial, that isC ³ is equivariantly isomorphic toGxC ^3–dimG . This result follows directly from the case of the additive groupG=G _a and is shown to hold for quasi-algebraic actions as well. Connections with the classification of homogeneous affine varieties are discussed.Partially supported by NSF grant DMS 8420315     

Morse theory for the space of Higgs <Emphasis Type="Italic">G</Emphasis>–bundles

Fix a C ^∞ principal G–bundle E⁰_G{E^0_G} on a compact connected Riemann surface X, where G is a connected complex reductive linear algebraic group. We consider the gradient flow of the Yang–Mills–Higgs functional on the cotangent bundle of the space of all smooth connections on E⁰_G{E^0_G}. We prove that this flow preserves the subset of Higgs G–bundles, and, furthermore, the flow emanating from any point of this subset has a limit. Given a Higgs G–bundle, we identify the limit point of the integral curve passing through it. These generalize the results of the second named author on Higgs vector bundles.     

Correspondance de Jacquet–Langlands explicite II. Le cas de degréégal à la caractéristique résiduelle

Let F be a non-Archimedean locally compact field, and let p be its residual characteristic. Put G=GL _p (F) and let G ^′=D _×, where $D$ is a division algebra with centre F and of degree p ² over F. The Jacquet–Langlands correspondence is a bijection between the discrete series of G and that of G ^′. We describe this explicitly, in terms of Carayol's parametrization of these discrete series. Received: 25 November 1999     

A user friendly proof of Nagata's characterization of linearly reductive groups in positive characteristics

Let K be an algebraically closed field of positive characteristic p, and G be a linear algebraic group over K. We give a user friendly proof of Nagata's theorem that every finite-dimensional rational representation of G is completely reducible if and only if the connected component G ⁰ is a torus and p does not divide the index (G?:?G ⁰).     

On the connectivity of unit distance graphs

For a number fieldK , consider the graphG(K^d), whose vertices are elements ofK ^d, with an edge between any two points at (Euclidean) distance 1. We show thatG(K²) is not connected whileG(K^d) is connected ford 5. We also give necessary and sufficient conditions for the connectedness ofG(K³) andG(K⁴).     

On open subsemigroups of connected groups

It is well known that a cancellative semigroup can be embedded into a group if it satisfies “Ore’s condition” of being either left or right reversible. However Ore’s condition is by no means necessary, so it is natural to ask which subsemigroups of a group are left or right reversible, or satisfy a condition of a similar type. In the present paper we study this question on open subsemigroups of connected locally compact groups; we also show how to use concepts related with reversibility to prove assertions like the following: Suppose thatS is an open subsemigroup of a connected Lie groupG such that 1 ∈ . IfG is solvable or ifS is invariant thenS is connected andS determinesG uniquely; that is to say, ifS can be embedded as an open subsemigroup into a connected Lie groupG’ thenG’ is isomorphic withG. Examples show that there are non-connected open subsemigroupsS of Sl(2,R) with 1 ∈ and such that the uniqueness assertion fails. The author gratefully acknowledges the support he received from the Alexander von Humboldt Foundation during the time he prepared this paper.     

Cross-sections, quotients, and representation rings of semisimple algebraic groups

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny t:[^(G)] ? G \tau :\hat{G} \to G is bijective; this answers Grothendieck’s question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg’s theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G] ^G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G] ^G and that of the representation ring of G and answer two Grothendieck’s questions on constructing generating sets of k[G] ^G . We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T ⇢ G/T where T is a maximal torus of G and W the Weyl group.     

Moduli spaces of principal <Emphasis Type="Italic">F</Emphasis>-bundles

In this paper we construct certain moduli spaces, which we call moduli spaces of (principal) F-bundles, and study their basic properties. These spaces are associated to triples consisting of a smooth projective geometrically connected curve over a finite field, a split reductive group G, and an irreducible algebraic representation .of of Our spaces generalize moduli spaces of F-sheaves, studied by Drinfeld and Lafforgue, which correspond to the case G = GL^r and is the tensor product of the standard representation and its dual. The importance of the moduli spaces of F-bundles is due to the belief that Langlands correspondence is realized in their cohomology.     

Commutative rings whose zero-divisor graph is a proper refinement of a star graph

A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G _c ^* be the subgraph of G induced on the vertex set V (G)\ {c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G = Γ(R), which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that G _c ^* has at least two connected components. We prove that the diameter of the induced graph G _c ^* is two if Z(R)² ≠ {0}, Z(R)³ = {0} and G _c ^* is connected. We determine the structure of R which has two distinct nonadjacent vertices α, β ∈ Z(R)* \ {c} such that the ideal [N(α) ∩ N(β)]∪ {0} is generated by only one element of Z(R)*\{c}. We also completely determine the correspondence between commutative rings and finite complete graphs K _n with some end vertices adjacent to a single vertex of K _n .     

A Note on Normality of Cones Over Symmetric Varieties

Let G be a semisimple and simply connected algebraic group, and let H ⁰ be the subgroup of points fixed by an involution of G. Let V be an irreducible representation of G with a nonzero vector v fixed by H ⁰. In this article, we prove a property of the normalization of the coordinate ring of the closure of G·[v] in ?(V).     

Equivariant K-Theory of Compact Connected Lie Groups Dedicated to Daniel Quillen on the Occasion of his Sixtieth Birthday

We compute the equivariant K-theory K _G ^* (G)for a compact connected Lie group Gsuch that ₁ (G)is torsion free (where Gacts on itself by conjugation). We prove that K _G ^* (G)is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a compact connected Lie group Gwith ₁ (G)torsion, namely PSU(3), and compute the corresponding equivariant K-theory.     

The number of spanning trees in regular graphs

Let C(G) denote the number of spanning trees of a graph G. It is shown that there is a function ?(k) that tends to zero as k tends to infinity such that for every connected, k-regular simple graph G on n vertices C(G) = {k[1 ? δ(G)]}ⁿ. where 0 ≤ δ(G) ≤ ?(k).     

A Note on Tortrat Groups

A locally compact group G is called a Tortrat group if for any probability measure on G which is not idempotent, the closure of {gg ^–1 | gG} does not contain any idempotent measure. We show that a connected Lie group G is a Tortrat group if and only if for all gG all eigenvalues of Ad g are of absolute value 1. Together with well-known results this also implies that a connected locally compact group is a Tortrat group if and only if it is of polynomial growth.