Given a representation R from G to G L V, the

The dual vector space V dual or V star is the space hom from V to C of linear maps from V to C. Equivalently, if we think of V as a space of column vectors by picking a basis of V then we can think of V dual as the space of row vectors: given an element alpha in hom V, C, we get a row vector underline alpha as follows. Pick a basis of V so that elements can be written as column vectors v equals v_1 down to v_n. Then alpha of v equals alpha_1 v_1 plus dot dot dot plus alpha_n v_n for some coefficients alpha_i. Define underline alpha to be the row vector alpha_1 up to alpha_n (so alpha of v equals underline alpha matrix product with v).
Conversely, if you have a row vector w equals w_1 up to w_n then we get the linear map overline w in hom V, C, defined by overline w over v equals w_1 v_1 plus dot dot dot plus w_n v_n.

The dual representation is defined as follows. For each g in G, define R star g from V dual to V dual to be the map which sends a row vector underline alpha to underline alpha matrix product with (R of g) inverse. In other words, underline (R star of g applied to alpha) equals alpha underline matrix product with (R of g) inverse.
The messing with underlines is just to distinguish between applying a linear map and multiplying a matrix, because we want to multiply on the right.