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Let G be a semisimple algebraic group defined over an algebraically closed field. We provide some criteria for normality and rational singularities of G-saturation under certain circumstances. Our results are applied to determine when the commuting variety over simple Lie algebra of low rank is normal and Cohen-Macaulay. We also present some interesting connections between injective modules and normality (resp. rational singularities) of their G-saturations. Finally, we generalize a machinery used to study singularities of nilpotent orbit closures.

In this two-part paper, we introduce a $$p$$ -adic analytic analogue of Backelin and Kremnizer’s construction of the quantum flag variety of a semisimple algebraic group, when $$q$$ is not a root of unity and $$\vert q-1\vert

For [math] a product of Severi–Brauer varieties, we conjecture that if the Chow ring of [math] is generated by Chern classes, then the canonical epimorphism from the Chow ring of [math] to the graded ring associated to the coniveau filtration of the Grothendieck ring of [math] is an isomorphism. We show this conjecture is equivalent to the condition that if [math] is a split semisimple algebraic group of type [math] , [math] is a Borel subgroup of [math] and [math] is a standard generic [math] -torsor, then the canonical epimorphism from the Chow ring of [math] to the graded ring associated with the coniveau filtration of the Grothendieck ring of [math] is an isomorphism. In certain cases we verify this conjecture.

Let $G$ be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic $p$, assumed to be larger than the Coxeter number. The "support variety" of a $G$-module $M$ is a certain closed subvariety of the nilpotent cone of $G$, defined in terms of cohomology for the first Frobenius kernel $G_1$. In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for $G = \mathrm{SL}_n$ in earlier work of the second author. In this paper, we show that for any $G$, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide (i.e., the Humphreys conjecture is true) when $p$ is sufficiently large. We also prove variants of these statements involving "relative support varieties.", 54 pages, 2 color figures. v3: minor corrections and additions

Let $G$ be a connected semisimple algebraic group and let $H \subset G$ be a connected reductive subgroup. Given a flag variety $X$ of $G$, a result of Vinberg and Kimelfeld asserts that $H$ acts spherically on $X$ if and only if for every irreducible representation $R$ of $G$ realized in the space of sections of a homogeneous line bundle on $X$ the restriction of $R$ to $H$ is multiplicity free. In this case, the information on restrictions to $H$ of all such irreducible representations of $G$ is encoded in a monoid, which we call the restricted branching monoid. In this paper, we review the cases of spherical actions on flag varieties of simple groups for which the restricted branching monoids are known (this includes the case where $H$ is a Levi subgroup of $G$) and compute the restricted branching monoids for all spherical actions on flag varieties that correspond to triples $(G,H,X)$ satisfying one of the following two conditions: (1) $G$ is simple and $H$ is a symmetric subgroup of $G$; (2) $G = \mathrm{SL}_n$., Comment: v3: 39 pages, some corrections in the list of references

This paper studies the “reduction mod p” method, which constructs large classes of representations for a semisimple algebraic group G from representations for the corresponding Lusztig quantum group U ζ at a p r -th root of unity. The G-modules arising in this way include the Weyl modules, the induced modules, and various reduced versions of these modules. We present a relation between Ext G n ( V , W ) and Ext U ζ n ( V ′ , W ′ ) , when V , W are obtained from V ′ , W ′ by reduction mod p. Since the dimensions of Ext n -spaces for U ζ -modules are known in many cases, our result guarantees the existence of many new extension classes and homomorphisms between rational G-modules. One application is a new proof of James Franklin's result on certain homomorphisms between two Weyl modules. We also provide some examples which show that the p-th root of unity case and a general p r -th root of unity case are essentially different.

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ be a finite-dimensional simple Lie superalgebra of type $D(2,1;\alpha)$, $G(3)$ or $F(4)$ over $\mathbb{C}$. Let $G$ be the simply connected semisimple algebraic group over $\mathbb{C}$ such that $\mathrm{Lie}(G)=\mathfrak{g}_{\bar{0}}$. Suppose $e\in\mathfrak{g}_{\bar{0}}$ is nilpotent. We describe the centralizer $\mathfrak{g}^{e}$ of $e$ in $\mathfrak{g}$ and its centre $\mathfrak{z}(\mathfrak{g}^{e})$ especially. We also determine the labelled Dynkin diagram for $e$. We prove theorems relating the dimension of $\left(\mathfrak{z}(\mathfrak{g}^{e})\right)^{G^{e}}$ and the labelled Dynkin diagram., Comment: 31 pages, 14 figures, accepted by Journal of algebra and its applications

Let H be a semisimple algebraic group, K a maximal compact subgroup of $$G:=H({{\mathbb {R}}})$$ G : = H ( R ) , and $$\Gamma \subset H({{\mathbb {Q}}})$$ Γ ⊂ H ( Q ) a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke–Maass forms on the locally symmetric space $$\Gamma \backslash G/K$$ Γ \ G / K to corresponding bounds on the arithmetic quotient $$\Gamma \backslash G$$ Γ \ G for cocompact lattices using the spectral function of an elliptic operator. The bounds obtained extend known subconvex bounds for automorphic forms to non-trivial K-types, yielding such bounds for new classes of automorphic representations. They constitute subconvex bounds for eigenfunctions on compact manifolds with both positive and negative sectional curvature. We also obtain new subconvex bounds for holomorphic modular forms in the weight aspect.

Let $G$ be a semisimple algebraic group with Lie algebra $\mathfrak g$. For a nilpotent $G$-orbit $\mathcal O\subset\mathfrak g$, let $d_\mathcal O$ denote the maximal dimension of a subspace $V\subset \mathfrak g$ that is contained in the closure of $\mathcal O$. In this note, we prove that $d_\mathcal O \le \frac{1}{2}\dim\mathcal O$ and this upper bound is attained if and only if $\mathcal O$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V= \frac{1}{2}\dim\mathcal O$, then $V$ is the nilradical of a polarisation of $\mathcal O$. Every nilpotent orbit closure has a distinguished $B$-stable subspace constructed via an $\mathfrak{sl}_2$-triple, which is called the Dynkin ideal. We then characterise the nilpotent orbits $\mathcal O$ such that the Dynkin ideal (1) has the minimal dimension among all $B$-stable subspaces $\mathfrak c$ such that $\mathfrak c\cap\mathcal O$ is dense in $\mathfrak c$, or (2) is the only $B$-stable subspace $\mathfrak c$ such that $\mathfrak c\cap\mathcal O$ is dense in $\mathfrak c$., 23 pages, to appear in the "Journal of the Australian Math. Soc."

Horospheres for an action of a semisimple algebraic group G on an affine variety X are the generic orbits of a maximal unipotent subgroup U ⊂ G or, equivalently, the generic fibers of the categorical quotient of the variety X by the action of U, which is defined by the values of the highest weight functions. The remaining fibers of this quotient (which we call degenerate horospheres) for a certain class of spherical G-varieties containing all simply connected symmetric spaces are studied.

In this paper we discuss the existence of certain classes of cuspidal automorphic representations having non-zero Fourier coefficients for a general semisimple algebraic group $ G$ defined over a number field $ k$ such that its Archimedean group $ G_\infty $ is not compact. When $ G$ is quasi-split over $ k$, we obtain a result on existence of generic cuspidal automorphic representations which generalize results of Vignéras, Henniart, and Shahidi. We also discuss: (i) the existence of cuspidal automorphic forms with non-zero Fourier coefficients for congruence of subgroups of $ G_\infty $, and (ii) applications related to the work of Bushnell and Henniart on generalized Whittaker models.

Let X be the variety of Borel subgroups of a split semisimple algebraic group G over a field, twisted by a generic G-torsor. Conjecturally, the canonical epimorphism of the Chow ring $$\mathop {\mathrm {CH}}\nolimits X$$ onto the associated graded ring GK(X) of the topological filtration on the Grothendieck ring K(X) is an isomorphism. We prove the new cases $$G={\text {Spin}}(11)$$ and $$G={\text {Spin}}(12)$$ of this conjecture. On an equivalent note, we compute the Chow ring $$\mathop {\mathrm {CH}}\nolimits Y$$ of the highest orthogonal grassmannian Y for the generic 11- and 12-dimensional quadratic forms of trivial discriminant and Clifford invariant. In particular, we describe the torsion subgroup of the Chow group $$\mathop {\mathrm {CH}}\nolimits Y$$ and determine its order which is equal to $$16\;777\; 216$$ . On the other hand, we show that the Chow group $$\mathop {\mathrm {CH}}\nolimits _0Y$$ of 0-cycles on Y is torsion-free.

The paper concerns a certain subcategory of the category of representations for a semisimple algebraic group $G$ in characteristic $p$, which arise from the semisimple modules for the corresponding quantum group at a $p$-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for $G$ when the Lusztig character formula does not hold., Comment: accepted for Special Issue on Representation Theory in Sci China Math

We look at the centralizer in a semisimple algebraic group $G$ of a regular nilpotent element, and show that its closure in the wonderful compactification is isomorphic to the Peterson variety. It follows that the closure in the wonderful compactification of the centralizer $G^x$ of any regular element $x$ is isomorphic to the closure of a general $G^x$-orbit in the flag variety. We also give a description of the $G^e$-orbit structure of the Peterson variety., Final version, to appear in Represent. Theory

In this article we investigate ordinary and equivariant Rost motives. We provide an equivariant motivic decomposition of the variety of full flags of a split semisimple algebraic group G over a field, a motivic decomposition of E/B for a G-torsor E over a smooth base scheme, study torsion subgroup of the Chow group of E/B, define some equivariant Rost motives over a field and some ordinary Rost motives over a general base scheme, and relate equivariant Rost motives with classifying spaces of some algebraic groups.

We prove that the variety of complete flags for any semisimple algebraic group is rigid in any smooth family of Fano manifolds.

Let $G$ be a complex simply-laced semisimple algebraic group of rank $r$ and $B$ a Borel subgroup. Let $\mathbf i \in [r]^n$ be a word and let $\mathbf \ell = (\ell_1,\dots,\ell_n)$ be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to $\mathbf i$ and $\mathbf \ell$ called a twisted cube, whose lattice points encode the character of a $B$-representation. More precisely, lattice points in the twisted cube, counted with sign according to a certain density function, yields the character of the generalized Demazure module determined by $\mathbf i$ and $\mathbf \ell$. In recent work, the author and Harada described precisely when the Grossberg-Karshon twisted cube is untwisted, i.e., the twisted cube is a closed convex polytope, in the situation when the integer sequence $\mathbf \ell$ comes from a weight $\lambda$ of $G$. However, not every integer sequence $\mathbf \ell$ comes from a weight of $G$. In the present paper, we interpret untwistedness of Grossberg-Karshon twisted cubes associated to any word $\mathbf i$ and any integer sequence $\mathbf \ell$ using the combinatorics of $\mathbf i$ and $\mathbf \ell$. Indeed, we prove that the Grossberg-Karshon twisted cube is untwisted precisely when $\mathbf i$ is hesitant-jumping-$\mathbf \ell$-walk-avoiding., Comment: Keywords: Grossberg-Karshon twisted cubes, pattern avoidance, character formula, generalized Demazure modules. arXiv admin note: text overlap with arXiv:1407.8543

According to a 2018 preprint by Nobuaki Yagita, the conjecture on a relationship between [Formula: see text]- and Chow theories for a generically twisted flag variety of a split semisimple algebraic group [Formula: see text], due to the author, fails for [Formula: see text] the spinor group [Formula: see text]. Yagita’s tools include a Brown–Peterson version of algebraic cobordism, ordinary and connective Morava [Formula: see text]-theories, as well as Grothendieck motives related to various cohomology theories over fields of characteristic [Formula: see text]. We provide a proof using only the [Formula: see text]- and Chow theories themselves and extend the (slightly modified) example to arbitrary characteristic.

Let G be a simply connected, semisimple algebraic group of type B4 or D4 over an algebraically closed field of characteristic p > 0. We determine the characters of certain simple modules for these groups by calculating the composition factors of the Weyl modules.

A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible A 1 subgroups of exceptional algebraic groups G. Consequences are given concerning the representations of such subgroups on various G-modules: for example, the conjugacy classes of irreducible A 1 subgroups are determined by their composition factors on the adjoint module of G.

Let $G$ be a semisimple algebraic group over a field of characteristic $p > 0$. We prove that the dual Weyl modules for $G$ all have $p$-filtrations when $p$ is not too small. Moreover, we give applications of this theorem to $p^n$-filtrations for $n > 1$, to modules containing the Steinberg module as a tensor factor, and to the Donkin conjecture on modules having $p$-filtrations., minor revisions and updates

We introduce a rigid analytification of the quantized coordinate algebra of a semisimple algebraic group and a quantized Arens-Michael envelope of the enveloping algebra of the corresponding Lie algebra, working over a non-archimedean field and when $q$ is not a root of unity. We show that these analytic quantum groups are topological Hopf algebras and Fr\'echet-Stein algebras. We then introduce an analogue of the BGG category $\mathcal{O}$ for the quantum Arens-Michael envelope and show that it is equivalent to the category $\mathcal{O}$ for the corresponding quantized enveloping algebra., Comment: 36 pages, Section 5 and more references added

Let G be a connected, semisimple algebraic group over a field k whose characteristic is very good for G. In a canonical manner, one associates to a nilpotent element X is an element of Lie(G) a parabolic subgroup P - in characteristic zero, P may be described using an sl(2)-triple containing X; in general, P is the "instability parabolic" for X as in geometric invariant theory. In this setting, we are concerned with the center Z(C) of the centralizer C of X in G. Choose a Levi factor L of P, and write d for the dimension of the center Z(L). Finally, assume that the nilpotent element X is even. In this case, we can deform Lie(L) to Lie(C), and our deformation produces a d-dimensional subalgebra of Lie(Z(C)). Since Z(C) is a smooth group scheme, it follows that dim Z(C) >= d = dim Z(L). In fact, Lawther and Testerman have proved that dim Z(C) = dim Z(L). Despite only yielding a partial result, the interest in the method found in the present work is that it avoids the extensive case-checking carried out by Lawther and Testerman.

Let $G$ be a semisimple algebraic group over an algebraically closed field of characteristic $p \geq 0$. At the 1966 International Congress of Mathematicians in Moscow, Robert Steinberg conjectured that two elements $a, a' \in G$ are conjugate in $G$ if and only if $f(a)$ and $f(a')$ are conjugate in $\operatorname{GL}(V)$ for every rational irreducible representation $f: G \rightarrow \operatorname{GL}(V)$. Steinberg showed that the conjecture holds if $a$ and $a'$ are semisimple, and also proved the conjecture when $p = 0$. In this paper, we give a counterexample to Steinberg's conjecture. Specifically, we show that when $p = 2$ and $G$ is simple of type $C_5$, there exist two non-conjugate unipotent elements $u, u' \in G$ such that $f(u)$ and $f(u')$ are conjugate in $\operatorname{GL}(V)$ for every rational irreducible representation $f: G \rightarrow \operatorname{GL}(V)$., Comment: to appear in Transform. Groups

Let $\G$ be a semisimple algebraic group over a number field $K$, $\mathcal{S}$ a finite set of places of $K$, $K_\mathcal{S}$ the direct product of the completions $K_v, v \in \mathcal{S}$, and $\OO$ the ring of $\mathcal{S}$-integers of $K$. Let $G = \G(K_\mathcal{S})$, $\Gamma = \G(\OO)$ and $\pi:G \rightarrow G/\Gamma$ the quotient map. We describe the closures of the locally divergent orbits ${T\pi(g)}$ %in $G/\Gamma$ where $T$ is a maximal $K_\mathcal{S}$-split torus in $G$. If $\# S = 2$ then the closure $\overline{T\pi(g)}$ is a finite union of $T$-orbits stratified in terms of parabolic subgroups of $\G \times \G$ and, consequently, $\overline{T\pi(g)}$ is homogeneous (i.e., $\overline{T\pi(g)}= H\pi(g)$ for a subgroup $H$ of $G$) if and only if ${T\pi(g)}$ is closed. On the other hand, if $\# \mathcal{S} > 2$ and $K$ is not a $\mathrm{CM}$-field then $\overline{T\pi(g)}$ is homogeneous for $\G = \mathbf{SL}_{n}$ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple $K$-ranks for $\G \neq \mathbf{SL}_{n}$. As an application, we prove that $\overline{f(\OO^{n})} = K_{\mathcal{S}}$ for the class of non-rational locally $K$-decomposable homogeneous forms $f \in K_{\mathcal{S}}[x_1, \cdots, x_{n}]$., Comment: arXiv admin note: text overlap with arXiv:1502.02297

The probabilistic Waring problem for finite simple groups asks whether every word of the form $w_1w_2$, where $w_1$ and $w_2$ are non-trivial words in disjoint sets of variables, induces almost uniform distribution on finite simple groups with respect to the $L^1$ norm. Our first main result provides a positive solution to this problem. We also provide a geometric characterization of words inducing almost uniform distribution on finite simple groups of Lie type of bounded rank, and study related random walks. Our second main result concerns the probabilistic $L^{\infty}$ Waring problem for finite simple groups. We show that for every $l \ge 1$ there exists $N = N(l)$, such that if $w_1, \ldots , w_N$ are non-trivial words of length at most $l$ in pairwise disjoint sets of variables, then their product $w_1 \cdots w_N$ is almost uniform on finite simple groups with respect to the $L^{\infty}$ norm. The dependence of $N$ on $l$ is genuine. This result implies that, for every word $w = w_1 \cdots w_N$ as above, the word map induced by $w$ on a semisimple algebraic group over an arbitrary field is a flat morphism. Applications to representation varieties, subgroup growth, and random generation are also presented., Comment: 45 pages

We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space $G/\Gamma$ of a semisimple algebraic group $G$. We define two families of algebraic subvarieties of the associated partial flag variety $G/P$, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of $m\times n$ real matrices whose image is not contained in any subvariety coming from these two families, the Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner's theorem on measure rigidity for unipotent flows, and linearization technique., Comment: 33 Pages. Removed Section 7 from the earlier version

In this chapter, we discuss the representation theory of semisimple algebraic groups. We also sketch the construction of finite dimensional irreducible representations of semisimple algebraic groups. We further discuss the geometric realization of finite dimensional irreducible representations of a semisimple algebraic group (over ℂ).

Let G be a complex semisimple algebraic group. In 2006, Belkale-Kumar defined a new product ⊙0 on the cohomology group H*(G/P,C) of any projective G-homogeneous space G/P. Their definition uses the notion of Levi-movability for triples of Schubert varieties in G/P.

For each irreducible module Xi Nanhua defined an element which generated this module. We use this element to construct a certain basis for and then compute dim , determine its formal characters in this paper. In order to obtain faster speed we modify the algorithm to compute the irreducible characters.

We study the relationship between two sets of coordinates on a double Bruhat cell, the cluster variables introduced by Berenstein, Fomin, and Zelevinsky and the X -coordinates defined by the coweight parametrization of Fock and Goncharov. In these coordinates, we show that the generalized Chamber Ansatz of Fomin and Zelevinsky coincides with the cluster ensemble map, a canonical monomial transformation between the cluster variables and X -coordinates defined by a common exchange matrix. We prove this in the setting of an arbitrary symmetrizable Kac–Moody group, generalizing along the way many previous results on the double Bruhat cells of a semisimple algebraic group. In particular, we construct an upper cluster algebra structure on the coordinate ring of any double Bruhat cell in a symmetrizable Kac–Moody group, proving a conjecture of Berenstein, Fomin, and Zelevinsky.

We prove most of Lusztig’s conjectures on the canonical basis in homology of a Springer fiber. The conjectures predict that this basis controls numerics of representations of the Lie algebra of a semisimple algebraic group over an algebraically closed field of positive characteristic. We check this for almost all characteristics. To this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the Springer resolution. On the one hand, this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and Rumynin. On the other hand, it is compatible with the t-structure arising from an equivalence with the derived category of perverse sheaves on the affine flag variety of the Langlands dual group. This equivalence established by Arkhipov and the first author fits the framework of local geometric Langlands duality. The latter compatibility allows one to apply Frobenius purity theorem to deduce the desired properties of the basis. We expect the noncommutative counterpart of the Springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric Langlands duality., United States. Air Force Office of Scientific Research (Grant FA9550-08-1-0315), National Science Foundation (U.S.) (Grant DMS-0854764), National Science Foundation (U.S.) (Grant DMS-1102434)

Let $ G $ be a connected, simply connected semisimple algebraic group over the complex number field, and let $ K $ be the fixed point subgroup of an involutive automorphism of $ G $ so that $ (G, K) $ is a symmetric pair. We take parabolic subgroups $ P $ of $ G $ and $ Q $ of $ K $ respectively and consider the product of partial flag varieties $ G/P $ and $ K/Q $ with diagonal $ K $-action, which we call a \emph{double flag variety for symmetric pair}. It is said to be \emph{of finite type} if there are only finitely many $ K $-orbits on it. In this paper, we give a parametrization of $ K $-orbits on $ G/P \times K/Q $ in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of $ P \subset G $ or $ Q \subset K $ is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of $ K $-spherical flag varieties $ G/P $ and $ G $-spherical homogeneous spaces $ G/Q $., 47 pages, 3 tables; add all the details of the classification

For an arbitrary connected solvable spherical subgroup H of a connected semisimple algebraic group G we compute the group N_G(H), the normalizer of H in G. Thereby we complete a classification of all (not necessarily connected) solvable spherical subgroups in semisimple algebraic groups., Comment: v2: 12 pages, small corrections

Let G be a connected semisimple algebraic group over $k$, with Lie algebra $\g$. Let $\h$ be a subalgebra of $\g$. A simple finite-dimensional $\g$-module V is said to be $\h$-indecomposable if it cannot be written as a direct sum of two proper $\h$-submodules. We say that $\h$ is wide, if all simple finite-dimensional $\g$-modules are $\h$-indecomposable. Some very special examples of indecomposable modules and wide subalgebras appear recently in the literature. In this paper, we describe several large classes of wide subalgebras of $\g$ and initiate their systematic study. Our approach is based on the study of idempotents in the associative algebra of $\h$-invariant endomorphisms of V. We also discuss a relationship between wide subalgebras and epimorphic subgroups., 14 pages; final version, to appear in "Algebras & Repr. Theory"

Let \(G\) be a connected semisimple algebraic group with Lie algebra \(\mathfrak{g }\) and \(P\) a parabolic subgroup of \(G\) with \(\mathrm{Lie\, }P=\mathfrak{p }\). The parabolic contraction \(\mathfrak{q }\) of \(\mathfrak{g }\) is the semi-direct product of \(\mathfrak{p }\) and a \(\mathfrak{p }\)-module \(\mathfrak{g }/\mathfrak{p }\) regarded as an abelian ideal. We are interested in the polynomial invariants of the adjoint and coadjoint representations of \(\mathfrak{q }\). In the adjoint case, the algebra of invariants is easily described and it turns out to be a graded polynomial algebra. The coadjoint case is more complicated. Here we found a connection between symmetric invariants of \(\mathfrak{q }\) and symmetric invariants of centralisers \(\mathfrak{g }_e\subset \mathfrak{g }\), where \(e\in \mathfrak{g }\) is a Richardson element with polarisation \(\mathfrak{p }\). Using this connection and results of Panyushev et al. (J Algebra 313:343–391, 2007), we prove that the algebra of symmetric invariants of \(\mathfrak{q }\) is free for all parabolic subalgebras in types \(\mathbf A\) and \(\mathbf C\) and some parabolics in type \(\mathbf B\). This technique also applies to the minimal parabolic subalgebras in all types. For \(\mathfrak{p }=\mathfrak{b }\), a Borel subalgebra of \(\mathfrak{g }\), one gets a contraction of \(\mathfrak{g }\) recently introduced by Feigin (Selecta Math 18:513–537, 2012) and studied from invariant-theoretic point of view in our previous paper (Panyushev and Yakimova in Ann Inst Fourier 62(6):2053–2068, 2012).