arXiv subjects

Jean Kaboré

Publications and source records attributed to Jean Kaboré.

3 recordsLinked to original sources

arXiv2026

Lowest-weight modules over the spherical subalgebra of a rational Cherednik algebra

Let $W$ be a finite reflection group acting on a finite-dimensional complex vector space $\h$, let $H_c(W,\h)$ be the associated rational Cherednik algebra, and let $\Dc = eH_ce$ be its spherical subalgebra. Building on the framework of BÃžgvad and Källström for invariant differential operators, subsequently applied to symmetric groups, we show that, for generic parameter $c$, a Lie subalgebra $\fa\subset\Dc$ generated by basic invariants always generates $\Dc$ itself (via the Levasseur--Stafford theorem), and that, granted a mild compatibility hypothesis between the grading element and $\fa$, verified here in the rank-one case, the category of $\Dc$-submodules of the polynomial ring $B=\mathbb C[\h]$ is equivalent to the category of modules over a Cartan-type ring $R_\fa$ built from $\fa$. The ring $R_\fa$ surjects onto the more naively defined $ \displaystyle R_c = \Dc^0 \big/ \big(\Dc^0 \cap \Dc \Dc^-\big), $ and the two coincide whenever this surjection is injective, which we verify in the rank-one case; whether this holds in general, and in particular for the families treated below, is left open. As an application, we determine explicit generators of $\fa$ for the imprimitive complex reflection groups $G(m,1,n)$ ($e=1$), recovering the classical Coxeter group $B_n$ as a special case. For $D_n=G(2,2,n)$, where an extra generator obstruction arises for $e>1$, we resolve the generation problem for $n=3$ completely by exploiting the exceptional isomorphism $D_3 \cong A_3$, leaving the general case $n\geq4$ open.

math.RT↗

arXiv2026

Differential Operators on $G(r,n)$-Invariant Functions

We generalize known results on normalized symmetric coordinates and their dual differential operators, established for the symmetric group, to the complex monomial reflection group $G(r,n):=μ_r\wr S_n$ (with $G(1,n)=S_n$). The central tool, proved in detail, is a \emph{transfer principle}: the substitution $y_i=x_i^r$ identifies the invariant ring of $G(r,n)$ with that of $S_n$ and transports, term by term, the corresponding operators and coordinates into explicit rational objects in the original variables $x_i$. From this we deduce the $G(r,n)$-analogues of the known results for $S_n$ existence and uniqueness of the dual coordinates $U_k$, and a Weyl algebra structure localized at the discriminant of $G(r,n)$ with complete and self-contained proofs for the points that do not follow directly from the transfer (Leibniz rule, main theorem). We then treat the \emph{total diagonal}: its preimage splits into $r^{n-1}$ lines permuted transitively by the group, and there, unlike the transferred operators $Δ_i$, the \emph{raw} derivatives $\partial_{x_i}$ exhibit a phenomenon specific to $r\ge2$ that we describe completely via a Fa di Bruno-type structure formula. Finally, we give a closed formula for the constants of this structure formula (via Bell polynomials), completely resolve the degeneracy at an isolated point $x_i=0$ (the operator $Δ_i$ extends holomorphically there, with $Δ_iϕ=\partial_i^rϕ/r!$), and deduce from this a partial analogue of the description of the tangent space to the GIT quotient $\CC^n/G(r,n)$; the case of several coordinates vanishing simultaneously remains open and is precisely delineated. Full proofs of all new results are given in detail.

math.AG↗

arXiv2026

A uniform decomposition theorem for invariant differential operators on imprimitive complex reflection groups G(r,p,n)

{We study the module structure of the polynomial ring, localized at the discriminant, over the ring of differential operators on the ring of invariants of the imprimitive complex reflection group $G(r,p,n)$, describing its simple components with explicit generators given by the higher Specht polynomials of Ariki--Terasoma--Yamada. The proof rests on a Jacobian lemma computing the discriminant of $G(r,p,n)$, combined with a double-centralizer argument. As particular cases ($r=2$, $p=2$ or $p=1$) we recover, and considerably shorten, the known decomposition theorems for the real reflection groups $W(D_n)$ and $W(B_n)$; we also treat $G(r,r,n)$ and $G(r,1,n)$ explicitly, with worked examples ($D_2$, $D_3$, $B_2$) and their central idempotents. Finally, applying the Galois descent equivalence of categories of Nonkané to $G(r,p,n)$ for the first time gives a second, generator-free description of the simple summands as twisted invariants.}

math.RT↗