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 p In this paper, we prove the degenerations of Schubert varieties in a minusculeG/P, as well as the class of Kempf varieties in the flag varietySL(n)/B, to (normal) toric varieties. Well known wonderfulG-varieties are those of rank zero, namely the generalized flag varietiesG/P, those of rank one, classified in [A], and certain complete symmetric varieties described in [DP] such as the famous space of complete conics. In this paper we compute the cohomology with trivial coefficients for the Lie superalgebraspsl(n, n), p (n) andq(2n); we show that the cohomology ring ofq(2n+1) is of Krull dimension 1 and we calculate the ring forq(3) andq(5). As a corollary we obtain af·g·p·d·f subgroup of SLn(?) (n ≧ 3. More generally, we prove that if Γ is an irreducible arithmetic non-cocompact lattice in a higher rank group, then Γ containsf·g·p·d·f groups. In the last section we give an exposition of results, communicated to us by J.-P. We prove that the moduli space of mathematical instanton bundles on P3 with c2 = 5 is smooth. We compute the ring of ${\mbox{\rm SL}}(2,{\mbox{\bf R}})$-invariants in the ring of polynomial functions, ${\mathcal P}$, on ${\mathcal A}$. We show that the absolute invariants (i.e.,the ${\mbox{\rm GL}}(2, {\mbox{\bf R}})$-invariants in the field of fractions of ${\mathcal P}$) distinguish the isomorphism classes of 2-dimensional non-associative real division algebras. Let G be a simple algebraic group over the algebraically closed field k of characteristic p ≥ 0. In case p >amp;gt; 0, assume G is defined and split over the finite field of p elements Fp. Let q be a power of p and let G(q) be the finite group of Fq-rational points of G. Assume B is F-stable, so that U is also F-stable and U(q) is a Sylow p-subgroup of G(q). It is proved that for any prime $p\geqslant 5$ the group $G_2(p)$ is a quotient of $(2,3,7;2p) = \langle X,Y: X^2=Y^3=(XY)^7 =[X,Y]^{2p}=1 \rangle.$ Given integers n,d,e with $1 \leqslant e >amp;lt; \frac{d}{2},$ let $X \subseteq {\Bbb P}^{\binom{d+n}{d}-1}$ denote the locus of degree d hypersurfaces in ${\Bbb P}^n$ which are supported on two hyperplanes with multiplicities d-e and e. For a finite-dimensional representation $\rho: G \rightarrow \mathrm{GL}(M)$ of a group G, the diagonal action of G on $M^p,$ p-tuples of elements of M, is usually poorly understood. Let k be an algebraically closed field of characteristic p ≥ 0. This result is not true when char k = p >amp;gt; 0 even in the case where H is a torus. However, we show that the algebra of invariants is always the p-root closure of the algebra of polarized invariants. Let p be a prime and let V be a finite-dimensional vector space over the field $\mathbb{F}_p$.

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