Lower bounds for Maass forms on semisimple groups

Let $G$ be an anisotropic semisimple group over a totally real number field $F$. Suppose that $G$ is compact at all but one infinite place $v_{0}$. In addition, suppose that $G_{v_{0}}$ is $\mathbb{R}$-almost simple, not split, and has a Cartan involution defined over $F$. If $Y$ is a congruence arithmetic manifold of non-positive curvature associated with $G$, we prove that there exists a sequence of Laplace eigenfunctions on $Y$ whose sup norms grow like a power of the eigenvalue.

Harish-Chandra Modules over ℤ

This chapter shows that certain classes of Harish-Chandra modules have in a natural way a structure over ℤ. The Lie group is replaced by a split reductive group scheme G/ℤ, its Lie algebra is denoted by 𝖌ℤ. On the group scheme G/ℤ there is a Cartan involution 𝚯 that acts by t ↦ t −1 on the split maximal torus. The fixed points of G/ℤ under 𝚯 is a flat group scheme 𝒦/ℤ. A Harish-Chandra module over ℤ is a ℤ-module 𝒱 that comes with an action of the Lie algebra 𝖌ℤ, an action of the group scheme 𝒦, and some compatibility conditions is required between these two actions. Finally, 𝒦-finiteness is also required, which is that 𝒱 is a union of finitely generated ℤ modules 𝒱I that are 𝒦-invariant. The definitions imitate the definition of a Harish-Chandra modules over ℝ or over ℂ.

Superintegrability of Generalized Toda Models on Symmetric Spaces

In this paper we prove superintegrability of Hamiltonian systems generated by functions on $K\backslash G/K$, restricted to a symplectic leaf of the Poisson variety $G/K$, where $G$ is a simple Lie group with the standard Poisson Lie structure, and $K$ is its subgroup of fixed points with respect to the Cartan involution.

Irreducible representations of E theory

We construct the [Formula: see text] theory analogue of the particles that transform under the Poincaré group, that is, the irreducible representations of the semi-direct product of the Cartan involution subalgebra of [Formula: see text] with its vector representation. We show that one such irreducible representation has only the degrees of freedom of 11-dimensional supergravity. This representation is most easily discussed in the light cone formalism and we show that the duality relations found in [Formula: see text] theory take a particularly simple form in this formalism. We explain that the mysterious symmetries found recently in the light cone formulation of maximal supergravity theories are part of [Formula: see text]. We also argue that our familiar space–times have to be extended by additional coordinates when considering extended objects such as branes.

A compact imbedding of Riemannian Symmetric Spaces

Let G be a connected real semisimple Lie group with finite center and θ be a Cartan involution of G. Suppose that K is the maximal compact subgroup of G corresponding to the Cartan involution θ. The coset space X = G/K is then a Riemannian symmetric space. In this paper, by choosing the reduced root system Σ0 = {α ∈ Σ | 2α /∈ Σ; α 2 ∈/ Σ} insteads of the restricted root system Σ and using the action of the Weyl group, firstly we construct a compact real analytic manifold Xb 0 in which the Riemannian symmetric space G/K is realized as an open subset and that G acts analytically on it, then we consider the real analytic structure of Xb 0 induced from the real analytic srtucture of AbIR, the compactification of the corresponding vectorial part.

An E11 invariant gauge fixing

We consider the nonlinear realisation of the semi-direct product of [Formula: see text] and its vector representation which leads to a space-time with tangent group that is the Cartan involution invariant subalgebra of [Formula: see text]. We give an alternative derivation of the invariant tangent space metric that this space–time possesses and compute this metric at low levels in eleven, five and four dimensions. We show that one can gauge fix the nonlinear realisation in an [Formula: see text] invariant manner.

A new algorithm for producing quantum circuits using KAK decompositions

We provide a new algorithm that translates a unitary matrix into a quantum circuit according to the G=KAK theorem in Lie group theory. With our algorithm, any matrix decomposition corresponding to type-AIII KAK decompositions can be derived according to the given Cartan involution. Our algorithm contains, as its special cases, Cosine-Sine decomposition (CSD) and Khaneja-Glaser decomposition (KGD) in the sense that it derives the same quantum circuits as the ones obtained by them if we select suitable Cartan involutions and square root matrices. The selections of Cartan involutions for computing CSD and KGD will be shown explicitly. As an example, we show explicitly that our method can automatically reproduce the well-known efficient quantum circuit for the $n$-qubit quantum Fourier transform.

Note on the Khaneja Glaser decomposition

Recently, Vatan and Williams utilize a matrix decomposition of $SU(2^n)$ introduced by Khaneja and Glaser to produce {\tt CNOT}-efficient circuits for arbitrary three-qubit unitary evolutions. In this note, we place the Khaneja Glaser Decomposition ({\tt KGD}) in context as a $SU(2^n)=KAK$ decomposition by proving that its Cartan involution is type {\bf AIII}, given $n \geq 3$. The standard type {\bf AIII} involution produces the Cosine-Sine Decomposition (CSD), a well-known decomposition in numerical linear algebra which may be computed using mature, stable algorithms. In the course of our proof that the new decomposition is type {\bf AIII}, we further establish the following. Khaneja and Glaser allow for a particular degree of freedom, namely the choice of a commutative algebra $\mathfrak{a}$, in their construction. Let $\chi_1^n$ be a {\tt SWAP} gate applied on qubits $1$, $n$. Then $\chi_1^n v \chi_1^n=k_1\; a \; k_2$ is a KGD for $\mathfrak{a}=\mbox{span}_{\mathbb{R}} \{ \chi_1^n ( \ket{j}\bra{N-j-1} -\ket{N-j-1}\bra{j}) \chi_1^n \}$ if and only if $v=(\chi_1^n k_1 \chi_1^n) (\chi_1^n a \chi_1^n)(\chi_1^n k_2 \chi_1^n)$ is a CSD.

Lefschetz numbers and unitary groups

We give a formula for the Euler-Poincare characteristic of the fixed point set of the Cartan involution on the set of integral equivalence classes of integral unimodular hermitian forms, in terms of a product of special values of Riemann zeta functions and Dirichlet L-functions. This is done via reduction by Galois cohomology to a volume computation using the Tamagawa measure on the unitary groups.