K-energy on polarized compactifications of Lie groups

We prove that the asymptotic Assouad-Nagata dimension of a connected Lie group $G$ equipped with a left-invariant Riemannian metric coincides with its topological dimension of $G/C$ where $C$ is a maximal compact subgroup. To prove it we will compute the Assouad-Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad-Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometric to any cocompact lattice on a connected Lie group.

We are concerned with conditions under which a locally compact group $G$ has a maximal compact normal subgroup $K$ and whether or not $G/K$ is a Lie group. If $G$ has small compact normal subgroups $K$ such that $G/K$ is a Lie group, then $G$ is pro-Lie. If in $G$ there is a collection of closed normal subgroups $\{ {H_\alpha }\}$ such that $\cap {H_\alpha } = e$ and $G/{H_\alpha }$ is a Lie group for each $\alpha$, then $G$ is a residual Lie group. We determine conditions under which a residual Lie group is pro-Lie and give an example of a residual Lie group which is not embeddable in a pro-Lie group.

Spherical functions on a real semisimple Lie group. A method of reduction to the complex case

This paper establishes a functorial algebraic isomorphism between the moduli space BCps(Σ,G) of polystable principal G-bundles with prescribed monodromy on a punctured Riemann surface Σ of genus g≥2, for a complex reductive Lie group G, and the character variety MCK(Σ*,G) of representations of its fundamental group with relatively compact image. The dimension formula dimBCps(Σ,G)=2(g−1)dimC(G)+∑i=1kdimR(Ci), where C1,…,Ck are conjugacy classes in a maximal compact subgroup K⊂G, is derived for complex reductive Lie groups, and singularities are characterized as polystable bundles with non-trivial automorphism groups. As applications of the above geometric results to control theory, it is proved that topologically distinct polystable robotic navigation strategies around obstacles are classified by this character variety. The geometry of singular points in families of polystable control strategies is further investigated, revealing enhanced stability properties characterized by reduced tangent space dimensions arising from non-trivial automorphism groups.

Let G be a connected complex simple Lie group with maximal compact subgroup U. Let g be the Lie algebra of G, and X = G/U be the associated Riemannian globally symmetric space of type IV. We have constructed three types of arithmetic uniform lattices in G, say of type 1, type 2, and type 3 respectively. If g is not equal to b_n, n&gt;0, then for each 0<i>4), c_n (n &gt; 5), or f_4. To prove these, we have simplified Kac's description of finite order automorphisms of g with respect to a Chevalley basis of g. Also we have determined some orientation preserving group action on some subsymmetric spaces of X.</i>

The first obstacle in building a Geometric Quantization theory for nilpotent orbits of a real semisimple Lie group has been the lack of an invariant polarization. In order to generalize the Fock space construction of the quantum mechanical oscillator, a polarization of the symplectic orbit invariant under the maximal compact subgroup is required.

Given a discrete subgroup Γ of a connected real semisimple Lie group G with finite center there is a natural homomorphism $$j_\Gamma ^q:I_G^q \to {H^q}\left( {\Gamma ;c} \right)\quad \left( {q = 0,1, \ldots } \right),$$ (1) where I G q denotes the space of G-invariant harmonic q-forms on the symmetric space quotient X=G/K of G by a maximal compact subgroup K. If Γ is cocompact, this homomorphism is injective in all dimensions and the main objective of Matsushima in [19] is to give a range m(G), independent of Γ, in which j Γ q is also surjective. The main argument there is to show that if a certain quadratic form depending on q is positive non-degenerate, then any Γ-invariant harmonic q-form is automatically G-invariant. In [3], we proved similarly the existence of a range in which j Γ q is bijective when Γ is arithmetic, but not necessarily cocompact. There are three main steps to the proof: (i) The cohomology of Γ can be computed by using differential forms which satisfy a certain growth condition, “logarithmic growth,” at infinity; (ii) up to some range c(G), these forms are all square integrable; and (iii) use the fact, pointed out in [16] , that for q ≦ m(G), Matsushima’s arguments remain valid in the non-compact case for square integrable forms.

Geometric quantization of real minimal nilpotent orbits

Let G be a p-adic connected reductive algebraic group and K a maximal compact subgroup of G. In [4], Casselman obtained the explicit formula of zonal spherical functions on G with respect to K on the assumption that K is special. It is known (Bruhat and Tits [3]) that the affine root system of algebraic group which has good but not special maximal compact subgroup is A1 C2, or Bn (n &gt; 3), and all Bn-types can be realized by orthogonal groups. Here the assumption “good” is necessary for the Satake’s theory of spherical functions.

Many moduli spaces in complex algebraix geometry can be expressed as quotients, in the sense of Mumford’s geometric invariant theory [18], of nonsingular complex projective varieties X by actions of complex reductive groups G. Any such quotient can also be identified with a symplectic quotient (or Marsden-Weinstein reduction) of the variety X by a maximal compact subgroup K of the reductive group G [14], [18], [19]. This symplectic quotient is µ-1(0)/K, where µ : X → k* is a moment map for the action of K on X equipped with a suitable symplectic form.

Let D = G / V D = G/V be a non-classical period domain, where G G is a semi-simple real Lie group with a compact Cartan subgroup T T and maximal compact subgroup K K , and V V the centralizer of a torus in G G . If the symmetric space is Hermitian, assume also that G / V G/V does not fiber holomorphically over G / K G/K . Let Γ \Gamma be a co-compact, torsion-free lattice in G G . We prove that the complex manifold Γ ∖ G / V \Gamma \backslash G/V does not admit any Kähler metric.

Let G be a reductive algebraic group over Q and Γ⊂G(Q) an arithmetic subgroup. Let K∞⊂G(R) be a maximal compact subgroup. We study the asymptotic behavior of the counting functions of the cuspidal and residual spectrum, respectively, of the regular representation of G(R) in L2(Γ﹨G(R)) of a fixed K∞-type σ. A conjecture, which is due to Sarnak, states that the counting function of the cuspidal spectrum of type σ satisfies Weyl's law and the residual spectrum is of lower order growth. Using the Arthur trace formula we reduce the conjecture to a problem about L-functions occurring in the constant terms of Eisenstein series. If G satisfies property (L), introduced by Finis and Lapid, we establish the conjecture. This includes classical groups over a number field.

This third chapter presents a brief and somewhat sketchy introduction to the theory of unitary representations of reductive and semisimple Lie groups G. The basic fact for an irreducible unitary representation π of G on a Hilbert space ℋ, is that every irreducible representation к of a maximal compact subgroup K ⊂ G has multiplicity m(к, π| к) ≤ dim к. This yields up the infinitesimal character χπ: L(g)→ ℂ and the distribution character θπ: C c ∞ (G) → ℂ, and consequently the differential equations $$z({\theta _\pi }) = \,{\chi _\pi }(z){\theta _\pi }$$ for $$z \in L{\text{(g)}}$$ which are the starting point for serious harmonic analysis on G.

We consider a homogeneous fibration G/L → G/K, with symmetric fiber and base, where G is a compact connected semisimple Lie group and L has maximal rank in G. We suppose the base space G/K is isotropy irreducible and the fiber K/L is simply connected. We investigate the existence of G-invariant Einstein metrics on G/L such that the natural projection onto G/K is a Riemannian submersion with totally geodesic fibers. These spaces are divided in two types: the fiber K/L is isotropy irreducible or is the product of two irreducible symmetric spaces. We classify all the G-invariant Einstein metrics with totally geodesic fibers for the first type. For the second type, we classify all these metrics when G is an exceptional Lie group. If G is a classical Lie group we classify all such metrics which are the orthogonal sum of the normal metrics on the fiber and on the base or such that the restriction to the fiber is also Einstein.

Let G be a semisimple real Lie group of non-compact type, K a maximal compact subgroup and S G a semigroup with nonempty interior.We consider the ideal boundary (G/K) of the associated symmetric space and the flag manifolds G/P .We prove that the asymptotic imagewhere C is the maximal invariant control set for the action of S in the flag manifold G/P , with P a parabolic subgroup.The points that project over C are exactly the points of type in (Sx 0 ) (in the sense of the type of a cell in a Tits Building).