Various constructions for quantum groups have been generalized to ıquantum groups. Such generalization is called ı-program. In this paper, we fill one of parts in the ı-program. Namely, we provide an equivariant K-theory approach to ı-quantum groups associated to the Satake diagram in (1), which is the Langlands dual picture of that constructed in [BKLW14], where a geometric realization of the ı-quantum group is provided by using perverse sheaves. As an application of the main results, we prove Li's conjecture [L18] for the special cases with the satake diagram in (1).

Let v = v0+v1 be a Z 2 -graded (super) vector space with an even C *action and χ ∈ v * 0 be a fixed point of the induced action. In this paper we prove an equivariant Darboux-Weinstein theorem for the formal polynomial algebraŝ A = S[v0] ∧χ ⊗ (v1). We also give a quantum version of it. Let g = g0 + g1 be a Lie superalgebra of type I and e ∈ g0 be a nilpotent element. We use the equivariant quantum Darboux-Weinstein theorem to give a Poisson geometric realization of the finite W -superalgebra U(g, e) in sense of Losev. An indirect relation between U(g, e) and the finite W-algebra U(g0, e) is presented. Finally we use such a realization to study the finite dimensional irreducible modules of U(g, e).

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Let G be a complex linear algebraic group, g = Lie(G) its Lie algebra and e ∈ g a nilpotent element. Vust's theorem says that in case of G = GL(V ), the algebra End Ge (V ⊗d ), where G e ⊂ G is the stabilizer of e under the adjoint action, is generated by the image of the natural action of d-th symmetric group S d and the linear maps {1 ⊗(i−1) ⊗ e ⊗ 1 ⊗(d−i) |i = 1, . . . , d}.In this paper, we generalize this theorem to G = O(V ) and SP(V ) for nilpotent element e with G • e being normal. As an application, we study the higher Schur-Weyl duality in the sense of [BK2] for types B, C and D, which establishes a relationship between W -algebras and degenerate affine braid algebras.

Various constructions for quantum groups have been generalized to \imath -quantum groups. Such a generalization is called an \imath -program. In this paper, we fill one of the parts in the \imath -program. Namely, we provide an equivariant K-theory approach to \imath -quantum groups, which is the Langlands dual picture of that constructed in Bao et al. (Transform. Groups 23 (2018), 329–389), where a geometric realization of \imath -quantum groups is provided by using perverse sheaves. As an application of the main results, we prove Li’s conjecture (Li, Represent. Theory 23 (2019), 1–56) for special cases.