Linear subspaces of the intersection of two quadrics via Kuznetsov component

Let $Q_i(i=1,2)$ be $2g$ dimensional quadrics in ${\mathbb{P}}^{2g+1}$ and let $Y$ be the smooth intersection $Q_1\cap Q_2$. We associate the linear subspace in $Y$ with vector bundles on the hyperelliptic curve $C$ of genus $g$ by the left adjoint functor of $\mathrm{\Phi }:D^b(C)\to D^b(Y)$. As an application, we give a different proof of the classification of line bundles and stable bundles of rank $2$ on hyperelliptic curves given by Desale and Ramanan. When $g=3$, we show that the projection functor induces a closed embedding $\alpha :Y\to S{U}_C^s(4,h)$ into the moduli space of stable bundles on $C$ of rank $4$ of fixed determinant.