A Soergel bimodule (for $S_n$) is a bimodule over $R = \mathbb{Q}[x_1,\dots,x_n]$ which appears as a summand/grading shift of tensor products of the basic bimodules $$B_{i,i+1} = R \otimes_{i,i+1} R$$ where $\otimes_{i,i+1}$ means the tensor product over the subring of polynomials invariant under permuting $i$ and $i+1$. It follows immediately that every Soergel bimodule $M$ has the following properties:

(1) $M$ is free as a left module or as a right module, although not necessarily as a bimodule.

(2) $M$ commutes with invariant polynomials, in the sense that for every invariant polynomial $p \in \mathbb{Q}[x_1,\dots,x_n]$ and $m \in M$, we have $$ pm = mp. $$

I think they also have the following property:

(3) There is an invariant vector, an element $m_0 \in M$ so that $$ x_i m_0 = m_0 x_i $$ for every $i=1,\dots,n$.

Do these properties characterize Soergel bimodules? Without the third condition, you could have, for instance, a bimodule that just permuted the $x_i$: a one-dimensional module with a single generator $a$ as a right module, so that $ x_i a = a x_{\sigma(i)} $ for some permutation $\sigma$.

**Edit:** The natural generalization for a general Weyl group $W$ would be to replace the invariant polynomials in (2) by the polynomials that are invariant under $W$. Clearly all Soergel bimodules would still satisfy this generalization of (2).

Any references are welcome. If it's not known, I'll try to prove it.

**Edit:** Ben Webster gave a counterexample below. More generally, I'm still interested in some sort of intrinsic, elementary characterization.

`$S_n$`

behaves better than other Weyl groups and their coinvariant algebras in Soergel's set-up, so I wonder if there is any hope for a general characterization along the lines you propose for this special case? $\endgroup$ – Jim Humphreys Mar 26 '12 at 13:39