## Equivariant cohomology of infinite-dimensional Grassmannian and shifted Schur functions.(English)Zbl 1300.14025

Denote by $$\mathcal{H}:=L^2(S^1)$$ the Hilbert space and let $$\mathcal{H}^+$$ and $$\mathcal{H}^-$$ be closed subspaces of $$\mathcal{H}$$ spanned by $$\{z^j:j\geq0\}$$ and $$\{z^j:j<0\}$$. The Segel-Wilson version Sato Grassmannian $$\mathrm{Gr}(\mathcal{H})$$ is defined to be a space of all closed linear subspaces $$W\subset \mathcal{H}$$ such that the projection $$\pi_{\_}: W\to \mathcal{H}_{\_}$$ is a Fredholm operator and the projection $$\pi_+: W\to \mathcal{H}_{+}$$ is a compact operator.
The group $$S^1=\{\alpha\in \mathbb{C}: |\alpha|=1\}$$ acts on $$\mathcal{H}$$ by assigning $$f$$ to $$\tilde{f}$$, where $$\tilde{f}(z)=f(\alpha z)$$. The action generates an action of $$S^1$$ on Grassmannian. If we represent a function on $$S^1$$ as a Fourier series $$f(z)=\sum a_n z^n$$, then this action sends $$a_k\to \alpha^k a_k$$. More general action of $$S^1$$ on $$\mathcal{H}$$ is given by sending $$a_k\to \alpha^{n_k}a_k$$ where $$n_k$$ is an arbitrary doubly infinite sequence of integers. Similarly, this action also generates an action of $$S^1$$ on Grassmannian $$\mathrm{Gr}(\mathcal{H})$$, which is a disjoint union of connected components $$\mathrm{Gr}_d(\mathcal{H})$$ labeled by the index of the projection $$\pi_{\_}: W\to \mathcal{H}_{\_}$$. All components are homeomorphic. Moreover, every component is homotopy equivalent to a subspace having a cell decomposition $$K=\cup \sigma_{\lambda}$$ consisting of even-dimensional cells. The cells are labeled by partitions. This decomposition is $$S^1$$-invariant and so the equivariant cohomology $$H_{S^1}(\mathrm{Gr}_d(\mathcal{H}))$$ has a free system of generators $$\Omega^T_{\lambda}:=[\overline{\Sigma}_{\lambda}]$$ as a module over $$H_{S^1}(pt)$$. These generators is interpreted as cohomology classes dual to Schubert cycles $$\overline{\Sigma}_{\lambda}$$ having finite codimension. In this paper, the main result is the calculation on the multiplication table in the basis $$\Omega^T_{\lambda}$$. The explicit formula is too complicated to state here.

### MSC:

 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 05E05 Symmetric functions and generalizations 14M15 Grassmannians, Schubert varieties, flag manifolds
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