Harish-Chandra highest weight representations of semisimple Lie algebras and Lie groups
In this expository paper we describe the theory of Harish-Chandra highest weight representations and their explicit geometric realizations.
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arXiv subjects
V. S. Varadarajan
Publications and source records attributed to V. S. Varadarajan.
Unitary Harish-Chandra representations of real supergroups
We give conditions for unitarizability of Harish-Chandra super modules for Lie supergroups and superalgebras.
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Highest weight Harish-Chandra supermodules and their geometric realizations
In this paper we discuss the highest weight $\frak k_r$-finite representations of the pair $(\frak g_r,\frak k_r)$ consisting of $\frak g_r$, a real form of a complex basic Lie superalgebra of classical type $\frak g$ (${\frak g}\neq A(n,n)$), and the maximal compact subalgebra $\frak k_r$ of $\frak g_{r,0}$, together with their geometric global realizations. These representations occur, as in the ordinary setting, in the superspaces of sections of holomorphic super vector bundles on the associated Hermitian superspaces $G_r/K_r$.
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Super Bundles
In this paper we give a brief account of the main aspects of the theory of associated and principal super bundles. As an application, we review the Borel-Weil-Bott Theorem in the super setting, and some results on projective embeddings of homogeneous spaces.
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Srishti Dhar Chatterji (1935-2017): In Memoriam
This article discusses the life and work of Professor Srishti Dhar Chatterji, who passed away on September 28, 2017, in Lausanne, Switzerland, most suddenly and unexpectedly, after a very brief illness. Complete bibliographical information is included
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Highest weight Harish-Chandra supermodules and their geometric realizations. I. The infinitesimal theory
In this series of papers we want to discuss the highest weight ${\frak k}_r$-finite representations of the pair $({\frak g}_r,{\frak k}_r)$ consisting of ${\frak g}_r$, a real form of a complex basic Lie superalgebra of classical type ${\frak g}$ (${\frak g}\neq A(n,n)$), and the maximal compact subalgebra ${\frak k}_r$ of ${\frak} g_{r,0}$. These representations will be concretely realized through spaces of sections of holomorphic vector bundles on the associated Hermitian superspaces. In this part we shall discuss only the infinitesimal theory of the pair $({\frak g}_r, {\frak k}_r)$. We treat the global theory in subsequent papers of the series.
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Temperedness of measures defined by polynomial equations over local fields
We investigate the asymptotic growth of the canonical measures on the fibers of morphisms between vector spaces over local fields of arbitrary characteristic. For non-archimedean local fields we use a version of the Łojasiewicz inequality (\cite{lojasiewicz1959}, \cite{hormander1958division}) which follows from Greenberg \cite{greenberg1966rational}, \cite{bollaerts1990estimate}, together with the theory of the Brauer group of local fields to construct definite forms of arbitrarily high degree, and to transfer questions at infinity to questions near the origin. We then use these to generalize results of H{ö}rmander \cite{hormander1958division} on estimating the growth of polynomials at infinity in terms of the distance to their zero loci. Specifically, when a fiber corresponds to a non-critical value which is stable, i.e. remains non-critical under small perturbations, we show that the canonical measure on the fiber is tempered, which generalizes results of Igusa and Raghavan \cite{igusa1978lectures}, and Virtanen and Weisbart \cite{virtanen2014elementary}.
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Structure, classification, and conformal symmetry of elementary particles over non-archimedean space-time
It is well known that at distances shorter than Planck length, no length measurements are possible. The Volovich hypothesis asserts that at sub-Planckian distances and times, spacetime itself has a non-Archimedean geometry. We discuss the structure of elementary particles, their classification, and their conformal symmetry under this hypothesis. Specifically, we investigate the projective representations of the $p$-adic Poincaré and Galilean groups, using a new variant of the Mackey machine for projective unitary representations of semidirect products of locally compact and second countable (lcsc) groups. We construct the conformal spacetime over $p$-adic fields and discuss the imbedding of the $p$-adic Poincaré group into the $p$-adic conformal group. Finally, we show that the massive and so called eventually masssive particles of the Poincaré group do not have conformal symmetry. The whole picture bears a close resemblance to what happens over the field of real numbers, but with some significant variations.
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Special geometry for arbitrary signatures
In this paper we generalize special geometry to arbitrary signatures in target space. We formulate the definitions in a precise mathematical setting and give a translation to the coordinate formalism used in physics. For the projective case, we first discuss in detail projective Kaehler manifolds, appearing in N=1 supergravity. We develop a new point of view based on the intrinsic construction of the line bundle. The topological properties are then derived and the Levi-Civita connection in the projective manifold is obtained as a particular projection of a Levi-Civita connection in a `mother' manifold with one extra complex dimension. The origin of this approach is in the superconformal formalism of physics, which is also explained in detail. Finally, we specialize these results to projective special Kaehler manifolds and provide explicit examples with different choices of signature.
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Structure, classifcation, and conformal symmetry, of elementary particles over non-archimedean space-time
It is known that no length or time measurements are possible in sub-Planckian regions of spacetime. The Volovich hypothesis postulates that the micro-geometry of spacetime may therefore be assumed to be non-archimedean. In this letter, the consequences of this hypothesis for the structure, classification, and conformal symmetry of elementary particles, when spacetime is a flat space over a non-archimedean field such as the $p$-adic numbers, is explored. Both the Poincaré and Galilean groups are treated. The results are based on a new variant of the Mackey machine for projective unitary representations of semidirect product groups which are locally compact and second countable. Conformal spacetime is constructed over $p$-adic fields and the impossibility of conformal symmetry of massive and eventually massive particles is proved.
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Airy functions over local fields
Airy integrals are very classical but in recent years they have been generalized to higher dimensions and these generalizations have proved to be very useful in studying the topology of the moduli spaces of curves. We study a natural generalization of these integrals when the ground field is a non-archimedean local field such as the field of p-adic numbers. We prove that the p-adic Airy integrals are locally constant functions of moderate growth and present evidence that the Airy integrals associated to compact p-adic Lie groups also have these properties.
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Airy Functions for Compact Lie Groups
The classical Airy function has been generalised by Kontsevich to a function of a matrix argument, which is an integral over the space of (skew) hermitian matrices of a unitary-invariant exponential kernel. In this paper, the Kontsevich integral is generalised to integrals over the Lie algebra of an arbitrary connected compact Lie group, using exponential kernels invariant under the group. The (real) polynomial defining this kernel is said to have the Airy property if the integral defines a function of moderate growth. A general sufficient criterion for a polynomial to have the Airy property is given. It is shown that an invariant polynomial on the Lie algebra has the Airy property if its restriction to a Cartan subalgebra has the Airy property. This result is used to evaluate these invariant integrals completely and explicitly on the hermitian matrices, obtaining formulae that contain those of Kontsevich as special cases.
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The Minkowski and conformal superspaces
We define complex Minkowski superspace in 4 dimensions as the big cell inside a complex flag supermanifold. The complex conformal supergroup acts naturally on this super flag, allowing us to interpret it as the conformal compactification of complex Minkowski superspace. We then consider real Minkowski superspace as a suitable real form of the complex version. Our methods are group theoretic, based on the real conformal supergroup and its Lie superalgebra.
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On the characterization of isotropic Gaussian fields on homogeneous spaces of compact groups
Let T be a random field invariant under the action of a compact group G We give conditions ensuring that independence of the random Fourier coefficients is equivalent to Gaussianity. As a consequence, in general it is not possible to simulate a non-Gaussian invariant random field through its Fourier expansion using independent coefficients.
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Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles
It is well known that the category of super Lie groups (SLG) is equivalent to the category of super Harish-Chandra pairs (SHCP). Using this equivalence, we define the category of unitary representations (UR's) of a super Lie group. We give an extension of the classical inducing construction and Mackey imprimitivity theorem to this setting. We use our results to classify the irreducible unitary representations of semidirect products of super translation groups by classical Lie groups, in particular of the super Poincaré groups in arbitrary dimension. Finally we compare our results with those in the physical literature on the structure and classification of super multiplets.
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Models for the irreducible representation of a Heisenberg group
In its most general formulation a quantum kinematical system is described by a Heisenberg group; the "configuration space" in this case corresponds to a maximal isotropic subgroup. We study irreducible models for Heisenberg groups based on compact maximal isotropic subgroups. It is shown that if the Heisenberg group is 2-regular, but the subgroup is not, the "vacuum sector" of the irreducible representation exhibits a fermionic structure. This will be the case, for instance, in a quantum mechanical model based on the 2-adic numbers with a suitably chosen isotropic subgroup. The formulation in terms of Heisenberg groups allows a uniform treatment of p-adic quantum systems for all primes p, and includes the possibility of treating adelic systems.
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On the deformation quantization of affine algebraic varieties
We compute an explicit algebraic deformation quantization for an affine Poisson variety described by an ideal in a polynomial ring, and inheriting its Poisson structure from the ambient space.
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Spinor Algebras and Extended Superconformal Algebras
We consider supersymmetry algebras in arbitrary spacetime dimension and signature. Minimal and maximal superalgebras are given for single and extended supersymmetry. It is seen that the supersymmetric extensions are uniquely determined by the properties of the spinor representation, which depend on the dimension $D$ mod 8 and the signature $|ρ|$ mod 8 of spacetime.
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