As a recently emerging technology, index-based modulation (IBM) has been attracting increasing research interests for its improved bit error rate (BER) performance and power efficiency. At present, the applications of two typical schemes named spatial modulation (SM) and subcarrier index modulation (IM), as well as their variants are introduced to microwave systems. To make IBM applicable in mmWave systems, the special properties of channel environments and system architectures should be taken into account. In this paper, we present a novel IBM scheme termed as generalized beamspace modulation (GBM) for mmWave beamspace multiple-input multiple-output (MIMO) systems. Unlike the frequency or spatial domain in which the existing IBM schemes are typically performed, GBM is implemented in the beamspace. To achieve near- optimal BER performance in GBM systems, a general effective beamspace channel (EBC) optimization method is derived based on the minimum asymptotic pairwise error probability (APEP) criterion. The optimal maximum-likelihood (ML) detector and the lowcomplexity detector are both provided. Thanks to our proposed GBM scheme, the BER performance can be noticeably enhanced compared to plain mmWave systems, with a smaller number of active frequency chains are used during transmission.