A Feynman-Kac formula for geometric quantization *

Reductive homogeneous Lorentzian manifolds

Let [Formula: see text] be a compact Lie group acting on a smooth manifold [Formula: see text]. In this paper, we consider Meinrenken’s [Formula: see text]-equivariant bundle gerbe connections on [Formula: see text] as objects in a 2-groupoid. We prove this 2-category is equivalent to the 2-groupoid of gerbe connections on the differential quotient stack associated to [Formula: see text], and isomorphism classes of [Formula: see text]-equivariant gerbe connections are classified by degree 3 differential equivariant cohomology. Finally, we consider the existence and uniqueness of conjugation-equivariant gerbe connections on compact semisimple Lie groups.

Functions generated by blocks were introduced by M. Taibleson and G. Weiss in the setting of the one-dimensional torus T T [TW1]. They showed that these functions formed a space "close" to the class of integrable functions for which we have almost everywhere convergence of Fourier series. Together with S. Lu [LTW] they extended the theory to the n n -dimensional torus where this convergence result (for Bochner-Riesz means at the critical index) is valid provided we also restrict ourselves to L log ⁡ L L\log L . In this paper we show that this restriction is not needed if the underlying domain is a compact semisimple Lie group (or certain more general spaces of a homogeneous type). Other considerations (for example, these spaces form an interesting family of quasi-Banach spaces; they are connected with the notion of entropy) guide one in their study. We show how this point of view can be exploited in the setting of more general underlying domains.

An efficient construction of lattice points FM of any density and any admissible symmetry is described in a finite region F⊃FM of a real n‐dimensional Euclidean space. The shape of F and the lattice symmetry of FM is determined by a compact semisimple Lie group of rank n underlying the construction. The density of FM is fixed by our choice of a positive integer M, where 1⩽M<∞. The Lie group allows one to introduce systems of special functions discretely orthogonal on FM together with the possibility of expanding digital data on FM into finite series of such special functions.

Flat connections on punctured surfaces and geodesic polygons in a Lie group

The Bott–Borel–Weil theorem (BBW) is a statement about a complex homogeneous space X=G/R where G is a compact semisimple Lie group and R is the centralizer of a torus in G. One knows that BBW is equivalent to the determination of how R operates on the cohomology of a certain nilpotent Lie algebra of antiholomorphic tangent vectors operating on an arbitrary irreducible G-module. Upon replacing the complex structure with a space S of spinors BBW is equivalent to a statement about the eigenvalues of the Casimir operator of R in the module S ⊗ Vλ where Vλ is the irreducible G-module with highest weight λ. But then the complex structure is eliminated and the problem makes sense for any compact subgroup R ⊂ G. We solve the problem in the case where R and G have the same rank. The very special case where R is the centralizer of a torus then yields BBW. Involved in the general case (i.e. arbitrary R) is a new (at least to mathematicians) Dirac operator with a cubic term whose square is expressed in terms of the Casimir operator. Actually the spectral resolution of the Dirac operator is determined in the general case. The kernel of the Dirac operator is only given in the equal rank case. A key feature in that case is that, associated to λ, there is a multiplet of representations of R, having rather striking properties, and cardinality equal to the Euler number of G/R.

Stability results for approximation by positive definite functions on SO(3)

If K is a connected compact Lie group with simple Lie algebra and if k is an integer relatively prime to the order of the Weyl group W of K then the number ν ( K , k ) \nu (K,k) of conjugacy classes of K consisting of elements x satisfying x k = 1 {x^k} = 1 is given by \[ ν ( K , k ) = ∏ i = 1 l m i + k m i + 1 , \nu (K,k) = \prod \limits _{i = 1}^l {\frac {{{m_i} + k}}{{{m_i} + 1}},} \] where l is the rank of K and m 1 , … , m l {m_1}, \ldots ,{m_l} are the exponents of W. If G is the complexification of K then we have ν ( G , k ) = ν ( K , k ) \nu (G,k) = \nu (K,k) without any restriction on k.

Sharp Fourier type and cotype of Lebesgue spaces and Schatten classes with respect to an arbitrary compact semisimple Lie group are investigated. In the process, a local variant of the Hausdorff-Young inequality on such groups is given.

Pluriclosed metrics on compact semisimple Lie groups

Let G be a compact semisimple Lie group, g its Lie algebra, (,) an AdG-invariant inner product on g, and M an adjoint orbit in 9. In this article, if (M,(,)|M) is Kähler with respect to its canonical complex structure, then we give, for a closed minimal Lagrangian submanifold L⊂M, upper bounds on the first positive eigenvalue λ1(L) of the Laplacian ΔL, which acts on C∞(L), and lower bounds on the volume of L. In particular, when (M,(,)|M) is Kähler-Einstein, (p=cω, where p and ω are Ricci form and Kähler form of (M,(,)|M) with respect to the canonical complex structure respectively, and c is a positive constant,) we prove λ1(L)≤c. Combining with a result of Oh [5], we can see that L is Hamiltonian stable if and only if λ1(L)=c.

A flag manifold is a homogeneous space M = G/K, where G is a compact semisimple Lie group, and K the centralizer of a torus in G. Equivalently, M can be identified with the adjoint orbit Ad (G)w of an element w in the Lie algebra of G. We present several aspects of flag manifolds, such as their classification in terms of painted Dynkin diagrams, T-roots and G-invariant metrics, and Kähler metrics. We give a Lie-theoretic expression of the Ricci tensor in M, hence reducing the Einstein equation on flag manifolds into an algebraic system of equations, which can be solved in several cases. A flag manifold is also a complex manifold, and this dual representation as a real and a complex manifold is related to a similar property of an infinite-dimensional manifold, the loop space, which in fact can be viewed as a "universal" flag manifold.

A generalized metric on a manifold M M , i.e., a pair ( g , H ) (g,H) , where g g is a Riemannian metric and H H a closed 3 3 -form, is a fixed point of the generalized Ricci flow if and only if ( g , H ) (g,H) is Bismut Ricci flat: H H is g g -harmonic and R c ( g ) = 1 4 H g 2 Rc(g)=\tfrac {1}{4}H_g^2 . On any homogeneous space M = G / K M=G/K , where G = G 1 × G 2 G=G_1\times G_2 is a compact semisimple Lie group with two simple factors, under some mild assumptions, we exhibit a Bismut Ricci flat G G -invariant generalized metric, which is proved to be unique among a 4 4 -parameter space of metrics in many cases, including when K K is neither abelian nor semisimple. On the other hand, if K K is simple and the standard metric is Einstein on both G 1 / π 1 ( K ) G_1/\pi _1(K) and G 2 / π 2 ( K ) G_2/\pi _2(K) , we give a one-parameter family of Bismut Ricci flat G G -invariant generalized metrics on G / K G/K and show that it is most likely pairwise non-homothetic by computing the ratio of Ricci eigenvalues. This is proved to be the case for every space of the form M = G × G / Δ K M=G\times G/\Delta K and for M 35 = S O ( 8 ) × S O ( 7 ) / G 2 M^{35}=SO(8)\times SO(7)/G_2 .

Kirillov’s character formula, the holomorphic Peter–Weyl theorem, and the Blattner–Kostant–Sternberg pairing

The compact simply connected Riemannian 4-symmetric spaces were classified by J. A. Jiménez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form G/H, where G is a connected compact simple Lie group with an automorphism γ˜ of order four on G and H is a fixed points subgroup Gγ of G. According to the classification by J. A. Jiménez, there exist seven compact simply connected Riemannian 4-symmetric spaces G/H in the case where G is of type E8. In the present article, we give the explicit form of automorphisms ω˜4, κ˜4 and ε˜4 of order four on E8 induced by the C-linear transformations ω4, κ4 and ε4 of the 248-dimensional C-vector space e8C, respectively. Further, we determine the structure of these fixed points subgroups (E8)ω4, (E8)κ4 and (E8)ε4 of E8. These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces G/H above corresponding to the Lie algebras h=su(2)⊕iR⊕e6, iR⊕so(14) and h=su(2)⊕iR⊕so(12), where h=Lie(H). With this article, the all realizations of inner automorphisms of order four and fixed points subgroups by them have been completed in E8.