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Let L be a finite-dimensional semisimple Lie algebra with a non-degenerate invariant bilinear form, sigma an elliptic automorphism of L leaving the form invariant, and A a sigma-invariant reductive subalgebra of L, such that the restriction of the form to A is non-degenerate. Consider the associated twisted affine Lie algebras L^, A^, and let F be the sigma-twisted Clifford module over A^ associated to the orthocomplement of A in L. Under suitable hypotheses onsigma and A, we provide a general formula for the decomposition of the kernel of the affine Dirac operator, acting on the tensor product of an integrable highest weight L^-module and F, into irreducible A^-submodules. As an application, we derive the decomposition of all level 1 integrable irreducible highest weight modules over orthogonal affine Lie algebras with respect to the affinization of the isotropy subalgebra of an arbitrary symmetric space.
Using representation theory, we compute the spectrum of the Dirac operator on the universal covering group of $SL_2(mathbb R)$, exhibiting it as the generator of $KK^1(mathbb C, mathfrak A)$, where $mathfrak A$ is the reduced $C^*$-algebra of the gro
The localized equivariant homology of the quiver variety of type $A_{N-1}^{(1)}$ can be identified with the level one Fock space by assigning a normalized torus fixed point basis to certain symmetric functions, Jack($mathfrak{gl}_N$) symmetric functi
We extend classical results of Kostant and al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting
We initiate the representation theory of the degenerate affine periplectic Brauer algebra on $n$ strands by constructing its finite-dimensional calibrated representations when $n=2$. We show that any such representation that is indecomposable and doe
The polynomial ring $B_r:=mathbb{Q}[e_1,ldots,e_r]$ in $r$ indeterminates is a representation of the Lie algebra of all the endomorphism of $mathbb{Q}[X]$ vanishing at powers $X^j$ for all but finitely many $j$. We determine a $B_r$-valued formal pow